UC-NRLF 


$B    27a    fiMfl 


>- 


T^f 


LX^  L 


C.  C 


IN   MEMORIAM 
FLORIAN  CAJORl 


/   ' 


u 


I  '^'^'  bj- 


u 


ELEMENTARY  ARITHMETIC; 


OR, 


SECOND     B.Q°Q,K 


SERIES    OF    MATHEMATICS, 


BY 

Andrew  "W.   Baker,  A.M.,  Ph.D., 

REVISED  AND  ENLARGED  EDITION. 


NEW    YORK: 
P.     O'SHEA,     PUBLISHER, 

37    BARCLAY     STREET. 
1880. 


■^32- 


Copyright,  1878,  1880,  by  P.  O'SBEA. 


CAJORl 


J.   CA.MPBKLI., 
PRINTER, 

15  Vandewator  St.,  N.  Y, 


PREFACE 


rpHIS  book  contains  the  Elements  of  Arithmetic,  or  perhaps 
more  properly  the  Elements  of  Mathematical  Science. 

In  its  preparation  I  have  endeavored  to  give  it  not  only 
a  mathematical  basis,  but  also  a  scientific  structure.  This  I 
have  done,  keeping  steadily  in  mind  that  it  is  a  book  for 
the  young,  for  whom  the  pathway  of  science  should  be  made 
as  easy  and  inviting  as  possible. 

We  cannot  insist  too  strongly  on  the  advantages  of  the 
blackboard  exercises  in  developing  the  principles  of  science, 
and  rendering  them  more  easily  and  more  thoroughly  under- 
stood. In  a  well-drilled  class,  whilst  one  student  is  at  the 
board  demonstrating  a  theorem  or  solving  a  problem,  and 
the  other  members  of  the  class  are  looking  on  with  attention, 
the  latter  learn  as  much  as  the  former. 

The  teacher  should  be  the  guide  of  the  class.  He  should 
not  be  satisfied  with  exemplification  only,  he  should  also 
endeavor  to  encourage  and  interest  his  pupils,  carefully 
observing  the  progress  made  each  day,  each  week,  each 
month,  and  at  the  end  of  the  year  when  all  shall  be  summed 
up,  such  will  be  the  delight  of  both  teacher  and  class  at 
the  progress  made,  that  they  will  begin  to  believe  there  is  a 
Toyal  road  to  learning. 


^ii:^ 


CONTENTS 


PAGK 

Definitions 7 

Mathematical  Terms 8 

Signs 9 

Axioms 9 

Notation  and  Numeration 10 

Addition  and  Subtraction 12 

Multiplication 21 

Division 2T 

Kbvibw 35 

Factoring 48 

Least  Common  Multiple 50 

Greatest  Common  Divisor 53 

Cancellation 53 

Fractions 56 

Addition  and   Subtraction  of 

Fractions 59 

Multiplication  of  Fractions...  61 

Division  of  Fractions 63 

Complex  Fractions. 66 

Review 68 

Decimal  Fractions 79 

Addition  and   Subtraction   of 

Decimal  Fractions 80 

Multiplication      of      Decimal 

Fractions 80 


FA«a 
Division     of     Decimal    Frac- 
tions      83 

Denominate  Numbers 83 

Simple  Ratio 111 

Compound  Ratio 117 

Percentage 119 

Commission  or  Brokerage 119 

Interest 123 

Bank  Discount 127 

Time  Discount 12i3 

Exchange 129 

Domestic  Exchange 129 

Foreign  Exchange 131 

Averaging  Accounts 132 

Bank  Account 133 

Alligation 134 

Alligation  Medial 134 

Alligation  Alternate 135 

Involution  and  Evolution 139 

Evolution 140 

Series     of     Common     Differ- 
ences    148 

General  Review 151 

Answers 165 


The  Author  cannot  too  highly  recommend  to 
the  Teacher  the  use  of  the  BlacJcboard  described 
on  the  following  page.  Great  facility  in  com- 
prehending the  combinations  and  divisions  of 
numbers  will  be  acquired  by  this  method. 


BLACKBOARD    EXERCISE. 

1  This  page  represents  a  blackboard  with  the  num-  37 

2  bers  as  high  as  72  painted  on  its  margins.  38 

3  There  is  also  a  box  containing  slips  which  will  cover  89 

4  two,  three,  four,  etc.,  as  high  as  12,  and  numbered  40 

5  accordingly  ;  one  of  these  the  student  will  take  in  his  41 

6  hand  and  apply  it  to  the  painted  numbers  to  perform  42 

7  addition  or  subtraction  ;  thus,  begin  at  1  and  take  a  43 

8  slip  marked  2,  then  1  and  2  are  3,  3  and  2  are  5, 5  and  44 

9  2  are  7,  7  and  2  are  9,  etc.,  counting  at  least  the  left-  45 

10  hand  column  ;  then,  to  perform  subtraction,  begin  at  46 

11  the  bottom  of  the  1st  column;  thus,  36  minus  2  equals  47 

12  34,  34-2=32,  32-2=30,  30-2=28,  etc.,  until  the  top  48 

13  is  reached  ;  then  taking  a  slip  marked  3,  begin  with  49 

14  1  or  2,  or  first  with  1  and  then  with  2,  and  return  to  50 

15  the  top  of  the  column  as  before,  by  subtraction  ;  let  51 

16  this  exercise  be  performed  with  all  the  slips,  and  as  52 

17  the  larger  numbers  are  taken,  continue  the  additions  53 

18  to  the  bottom  of  the  2d  column,  and  return  as  before.  54 

19  For  multiplication  and  division  first  make  a  chalk  55 

20  mark  after  every  two  figures  up  to  24,  and  mul-  56 

21  tiply ;  thus,  once  2  are  2,  twice  2  are  4,  3  times  2  are  57 

22  6,  4  times  2  are  8,  etc. ;  then  the  number  of  divisions  58 

23  is  12  and  each  division  has  2  numbers ;  .*.  12  is  con-  59 

24  tained  twice  in  24,  or  2  is  contained  12  times,  2  is  60 

25  contained  once  in  2,  in  4  twice,  in  6  three  times,  in  8  61 

26  four  times,  in  10  five  times,  in  12  six  times,  etc.    When  62 

27  the  student  is  familiar  with  multiplication  and  division  63 

28  by  2,  let  the  numbers  be  separated  into  3's,  then  4's,  64 

29  etc.,  and  let  each  be  continued  for  12  divisions  ;  when  65 

30  all  the  divisions  have  been  performed  according  to  the  66 

31  steps,  beginning  with  2  and  ending  with  12,  a  multi-  67 

32  plication  and  division  table  will  be  made.  68 

33  Rem. — In  multiplication  the  product  of  any  two  f  ac-  69 

34  tors  is  the  same  by  making  either  the  multiplicand  and  70 
OK  the  other  the  multiplier  ;  so  also  in  division,  the  divisor  «,^ 

and*  the  quotient  may  be  substituted,  as  the  dividend 

3^  is  the  product  of  the  divisor  and  quoti**nt.  *^ 
Rem. — The  numbers,  continued  up  W  144,  should 
be  painted  on  the  sides  of  the  board. 


ELEMEITARY  AiifHIETld 


■»»» 


Defij^itioj^s. 

1.  Arithmetic  is  the  science  of  numbers. 
3.  A  Unit  is  a  single  thing ;  as,  a  book,  one  dollar, 
or  simply  one. 

3.  A  Nwmhev  is  a  unit  or  a  collection  of  units ;  as, 
one,  ten,  five  books,  twenty-five  dollars. 

4.  The  numbers  used  in  Arithmetic  are  all  formed  by 
combinations  of  the  ten  Arabic  characters,  called  Fig- 
ures;  viz.,  0,  called  zero  or  naught;  1,  called  one; 
2,  two;  3,  three;  4,  four;  5,  five;  6,  six;  7,  seven; 
8,  eight ;  9,  nine. 

5.  Expressing  a  number  either  in  writing  or  figures  is 
called  dotation,  and  reading  the  expression  is  called 
Ifumeration. 

6*  When  numbers  are  used  without  reference  to  any 
object,  they  are  called  Abstract  Numbers  ;  as,  five, 
twenty,  etc. ;  but  when  they  are  applied  to  things,  they 
are  called  Concrete ;  as,  one  book,  ten  men,  four  dol- 
lars, etc. 

7.  When  concrete  numbers  express  values  of  money, 
weights,  measures,  time,  etc.,  they  are  called  Denomi" 


b  DEFINITIONS. 

nate  Numbers  ;  as,  dollars,  pounds,  shillings,  pounds 
of  weight,  ounces,  hours,  minutes,  etc. 

8.  When  different  denominations  of  either  kind  form 
burt  One  nUnibe'rv  .it^  i^  called  a  Compound  Number; 
as,  ^4  3s.  6d.;  ^  Ib/i  oz.  3  pwt.  and  2  gr. 

>'0v'^u.m'yejrs  6f>,.the  ^me  order  and  the  same  denom- 
ination are  termed  Like  Num^bers;  other  numbers 
are  termed  Unlike  Numbers. 

Rem. — Numbers  expressing  different  species  of  the  same  genus 
are  unlike,  as  horses  and  cows ;  while  the  same  numbers  expressed 
in  the  term  of  the  genus  are  alike,  as  animals. 

MATHEMATICAL  TEEMS  TJSED  IN  AEITH- 
METIC. 

1,  An  affirmative  sentence,  or  anything  proposed  for 
consideration,  is  a  Proposition. 

3.  A  self-evident  proposition  is  called  an  Axiom. 

3.  A  proposition  made  evident  by  a  demonstration  is 
called  a  Theorem. 

4.  When  a  proposition  is  used  for  developing  a  prin- 
ciple of  Arithmetic,  it  is  called  a  Problem* 

5.  Propositions  given  merely  for  solution,  in  order  to 
impress  the  principles  on  the  mind,  are  called  JExam^ 
pies. 

6.  An  obvious  consequence  of  one  or  more  proposi- 
tions is  called  a  Corollary. 

H.  An  established  custom,  or  an  assumption  without 
proof,  is  called  a  Postulate. 

Rem,  1  and  1  are  2,  3  and  1  are  3,  3  and  1  are  4,  5  and  2  are  7, 
6  and  3  are  9,  etc.,  is  the  postulate  which  forms  the  basis  of  Arith" 
metic. 


DEFINITIONS, 


AXIOMS, 


I.  If  equal  numbers  are  added  to  equal  numbers,  the 
sums  will  be  equal. 

3.  If  equal  numbers  are  subtracted  from  equal  num- 
bers, the  remainders  will  be  equal. 

3.  If  equals  be  multiplied  by  equals,  the  products  will 
be  equal. 

4.  If  equals  be  divided  by  equals,  the  quotients  will  be 
equal. 

5.  If  two  numbers  are  each  equal  to  the  same  number, 
they  are  equal  to  each  other. 

6.  If  the  same  number  be  added  to  and  subtracted 
from  another  number,  the  latter  number  will  not  be 
changed. 

?•  If  a  number  be  both  multiplied  and  divided  by  the 
same  number,  the  former  number  will  not  be  changed. 

8.  If  two  numbers  be  equally  increased  or  diminished, 
the  difference  of  the  resulting  numbers  will  be  the  same 
as  the  difference  of  the  originals. 

9.  If  two  numbers  are  like  parts  of  equal  numbers, 
they  are  equal  to  each  other. 

10.  The  whole  is  greater  than  any  of  its  parts. 

II.  The  whole  is  equal  to  the  sum  of  all  its  parts. 

SIGN^S. 

1.  The  sign  +,  called  plus^  is  the  sign  of  addition, 
and  indicates  that  the  number  on  the  right  hand  is  to  be 
added  to  the  one  on  the  left. 


10  NOTATION  AND    NUMERATION 

2.  The  sign  — ,  called  minuSj  is  the  sign  of  sub- 
traction, and  indicates  that  the  number  on  the  right  is 
to  be  subtracted  from  that  on  the  left. 

3.  The*  sign  x ,  called  into,  is  the  sign  of  multipli-^ 
cation,  and  indicates  that  the  numbers  between  which  it 
is  placed  are  Victors  of  the  same  product, 

4.  The  sign  -^,  divided  hy^  the  left-hand  number 
~  to  be  divided  by  the  right  hand. 

5.  The  sign  =,  equal  to^  indicates  that  the  num- 
bers between  which  it  is  placed  are  equal. 

6.  52,  53,  the  2  and  3  placed  to  the  right,  a  little 
above  a  number,  indicates  the  power  to  which  it  is  to 
be  raised. 

7.  V  ,  indicates  the  extraction  of  the  square  root ; 
and  \/~,  indicates  the  extraction  of  the  citbe  root. 


JYOTATIOJf    AJfD    J\[*UMERATIOJ^. 

1st.  A  figure  standing  alone,  as  1,  2,  3,  holds  the  units 
place,  or  is  of  the  1st  order,  and  is  read,  one,  two,  three, 

2d.  A  number  having  two  figures,  as  14,  26,  the  right- 
hand  figure  holds  the  units  place,  and  the  left-hand  figure 
that  of  tens,  and  they  are  read,  fourteen,  twenty-six. 

Cor. — The  right-hand  figure  of  a  number  is  called  units,  or  the 
1st  order ;  the  next  figure  to  the  left  is  called  tens^  or  the  2d 
order ;  the  third  figure,  hundreds,  or  the  3d  order ;  the  fourth 
figure,  thousands,  or  the  4th  order ;  and  if  a  number  be  expressed 
with  the  nine  figures  in  order,  making  1  the  right-hand  figure,  the 
figures  will  express  their  respective  orders  ;  thus. 


NOTATION  AND  NUMERATION.  11 

millions,  thousands,    units. 


m  m  nx 

•^  CM  "^  «Ht  'O  <M 

gcgO        g^O        g^O 

rtJoa         rtj^M  "rJOQ 

gcQ-M  goQ-M  goQ-tS 

r£3-Mf3       ,£^42?^      ^4S;S 

987,654,321 

If  pointed  in  periods  of  three  figures  each,  they  may  be  read  as 
follows  :  Nine  hundred  and  eighty-seven  millions  six  hundred  and 
fifty 'four  thousand  three  hundred  and  twenty-one. 

Rem. — The  figures  designate  the  orders. 

Bead  the  following  numbers: 

-L---------------   one, 

'^J-        -_----«------        twenty-oue, 

OiiiL        _       -       -       .       -       -       -       -       -  three  hundred  and  twenty-one. 

rr,0/v±        _._..._     four  thousand  three  hundred  and  twenty-one. 

04t,O/wi.        -_-_>-     fifty-four  thousand  three  hundred  and  twenty-one. 

Ut>'±,0/vX        ...  six  hundred  and  fifty-four  thousand  three  hundred  and  twenty- 

I  jDtlTjO/v'l.      seven  millions  six  hundred  and  fifty-four  thousand  three  hundred  and  twenty-one, 
07  £•«/ 091    J  eighty-seven  millions 
O  4  ,UU'±,0/Vi  \  six  hundred  and  fifty-four  thousand  three  hundred  and  twenty-one. 

QR*?  (\KA.  ^91    [  °^°®  hundred  and  eighty-seven  millions 

VO  i  y\JO'±^0/iil.  \  gix  hundred  and  fifty-four  thousand  three  hundred  and  twenty-one. 

Rem. — The  column  of  I's  is  of  the  1st  order,  the  column  of  2's 
is  of  the  2d  order,  the  3's  the  3d  order,  the  4's  the  4th  order,  etc. 

Cor. — The  relation  of  any  two  consecutive  orders  is  the  same, 
for  when  in  addition  the  sum  of  any  column  reaches  10,  the  left- 
hand  figure  belongs  to  the  next  column  or  order  ;  hence,  a  table 
may  be  formed,  thus, 

10  units  =  1  ten. 

10  tens  =  1  hundred. 

10  hundred  =  1  thousand. 

10  thousand  =  1  ten-thousand. 

10  ten- thousand  =  1  hundred-thousand. 

10  hundred-thousand  =  1  million. 

etc.  etc. 


jIdditioj^  Aj\rD  Subtbactio:n'. 


Addition  and  Subtkaction  Table. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

'1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

% 

3 

4 

5 

6 

7 

8 

9 

10 

11 

3 

•  4 

5 

6 

7 

S 

9 

10 

11 

12 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

9 
10 

10 
11 

11 
12 

12 
13 

13 

14 

15 
16 

16 

17 

18 

14 

15 

17 

18 

19 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

19 
20 

20 

21 

22 

23 

24 

24 

25 

26 

27 

28 

21 

22 

23 

25 

26 

27 

28 

29 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

ADDITION  AND    SUBTRACTION  13 

The  square  made  by  the  ten  Arabic  characters  forms 
an  Addition  and  Subtraction  Table. 

Beginning  with  the  first  line,  thus.  Zero  and  zero  are 
zero ;  zero  and  1  are  1 ;  zero  and  2  are  2 ;  zero  and  3 
are  3 ;  zero  and  4  are  4 ;  zero  and  5  are  5 ;  zero  and  6 
are  6,  etc. 

The  second  line,  one  and  zero  are  one  ;  1  and  1  are  2 : 

1  and  2  are  3  ;  1  and  3  are  4;  1  and  4  are  5,  etc. 

2  and  zero  are  2  ;  2  and  1  are  3 ;  2  and  2  are  4 ;  2  and 
3  are  5  ;  3  and  4  are  7,  etc. 

3  and  0  are  3 ;  3  and  1  are  4;  3  and  2  are  5 ;  3  and  3 
are  6 ;  3  and  4  are  7,  etc. 

Continue  this,  taking  the  first  figure  of  the  1st  column 
and  adding  it  to  each  successive  figure  in  the  first  line ; 
the  adding  of  zero  is  only  nominal,  as  it  makes  no 
increase. 

It  also  becomes  a  subtraction  table,  the  figures  of  the 
first  column  being  the  subtrahend,  and  those  of  the  first 
line  the  remainders. 

Take  zero  from  1  and  1  remains  ;  0  from  2,  2  remain, 
etc.  It  may  be  thus  expressed :  1— 0  =  1;  2  —  0  =  2; 
3  —  0  =  3;  4  —  0  =  4;  5  —  0  =  5;  which  is  read, 
1' minus  zero  equals  1,  etc. 

Second  line:  take  1  from  2,  1  remains;   1  from  3, 

2  remain ;    or,   2-^1  =  1;    3  —  1  =  2;    4  —  1  =  3; 
5  —  1  =  4,  etc. 

Third  line;  3-2  =  1;  4-2  =  2;  5-2=3; 
6—2  =  4;   7  —  2  =  5,  etc. 

In  addition,  we  add  two  numbers  at  a  time,  never  more,  and  in 
the  first  square  we  have  the  addition  of  every  two  units  that  can 
come  together  ;  so  also  in  subtraction. 


14  ADDITION   AND    SUBTRACTION, 

In  the  second  square,  the  units  correspond  with 'the  first  square^ 
and  have  an  additional  ten. 

In  the  third  square,  the  units  again  are  repeated,  and  another 
additional  ten. 

As  a  column  of  tens,  hundreds,  and  every  higher  or  lower 
order  is  added  and  subtracted  in  the  same  way,  the  above  table 
'leveiops  every  principle  of  addition  and  subtraction. 

Add  the  column  of  units. 

One  and  2  are  3  ;  3  and  3  are  6  ; 
6  and  4  are  10  ;  10  and  5  are  15  ;  15 
and  6  are  21 ;  21  and  7  are  28 ;  or  as 
is  customary  to  begin  at  the  bottom 
of  the  column,  7  and  6  are  13  ;  13 
and  5  are  18;  18  and  4  are  22;  22 
and  3  are  25;  25  and  2  are  27;  27 
and  1  are  28. 

9  +  7  =  16;  16  +  2  =  18;  18  + 
4  =  22 ;  22  +  6  =  28;  28  +  5  =  33 ;  33  +  3  =  36. 

Eem. — Although  many  numbers   may  be  added  together,  in  - 
performing  the  operation  only  two  at  a  time  are  added. 

Add  the  following  numbers  jointly  and  separately? 

thus, 

35 

24 

43 

52        = 

67        = 
221 

The  sum  of  the  column  of  units  is  21 ;  that  is,  1  unit 
and  2  tens ;  the  sum  of  the  column  of  tens  is  20;  that  is, 
20  tens  or  2  hundred;  and  the  two  sums  united  make 


1 

3 

3 

5 

3 

6 

4 

4 

5 

2 

6 

7 

2 

9 

28 

36 

30 

and  5 

20 

"   4 

40 

«   3 

50 

«  .  2 

21 

60 

"  A 

200 

200 

«  '21  = 

=  221 

ADDITION    AND    SUBTRACTION. 


15 


221 ;  precisely  the  same  as  if  the  column  of  units  is  first 
added,  and  the  units  of  the  sum  placed  under  the  column 
of  units,  and  the  tens  added  with  tiie  column  of  tens  ; 
and  then  the  tens  of  the  sum  of  the  tens  column  placed 
under  the  column  of  tens,  and  the  hundreds  in  place  of 
hundreds. 

CoE. — As  the  relation  of  each  successive  order  is  the 
same,  hence  for  every  ten  of  any  order,  the  1,  or  left- 
hand  figure,  belongs  to  the  next  order ;  and  the  process 
is  the  same  in  the  addition  of  every  column ;  that  is,  one 
is  carried  to  the  next  column  for  every  ten  in  the  addi- 
tion of  each  column. 


ADDITION-. 

STJBTEACTION". 

3241 

4365 

643315 

876432 

987654 

4356 

5331 

533684 

543210 

321334 

6745 

7546 

478921 

5364 

8432 

586432 

876543 

789654 

19706 

654331 

331043 

754331 

864331 

345678 

869754 

678643 

678963 

987654 

654321 

594721 

987654 

331987 

367543 

456789 

654331 

Add 

7654 

3897 

11551 

3465321 

6354789 

9830110 

Subtract 

Minuend,        11551 

Minuend, 

11551 

Subtrahend,      7654 

Subtrahend, 

3897 

3897 


7654 


16  ADDITION    AND    SUBTRACTION, 

Minuend,        9820110  Minuend,        9820110 

Subtrahend,    3465321  SuUraJiend,    6354789 

6354789  3465321 

Cor.  1. — The  minuend  is  always  equal  to  the  sum  of 
the  subtrahend  and  remainder,  and  is  therefore  greatei 
than  either. 

CoR.  2. — Arithmetic  is  based  upon  the  postulate 
contained  in  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  ivhich  is 
addition,  etc. ;  and  the  application  of  Axiom  6  (page  9) 
to  this  postulate  proves  the  principle  of  subtraction; 
thus,  19  +  6  =  25,  then  25—6  must  be  equal  to  19. 

Rem.  1.— Wlien  the  figure  of  the  subtrahend  is  larger  than 
the  one  above  it  of  the  same  order  of  the  minuend,  1  of  the  next 
order  of  the  minuend  must  be  united  to  the  figure  of  the  minuend 
and  then  the  subtraction  be  performed ;  then  in  order  to  make 
up  for  this  addition  to  the  minuend,  1  must  be  added  to  the  next 
order  of  the  subtrahend,  and  then  perform  the  subtraction  ;  this 
process  i-s  called  carrying  and  requires  all  the  attention  of  the 
student. 

Rem.  3. — I  prefer  few  examples,  but  these  may  be  often 
repeated,  and  if  thought  necessary  the  teacher  can  give  others  in 
which  the  columns  are  longer. 

Rem.  3.— Each  order  may  be  regarded  as  units,  and  the  sum 
may  reach  one,  two,  or  more  hundred  of  its  order. 

PRACTICAL     EXAMPLES 

1.  A  farmer  has  17  sheep  in  one  pasture,  41  in  another, 
and  57  in  a  third;  how  many  sheep  has  he  ? 

2.  A  fowler  has  18  turkeys,  21  geese,  42  ducks,  and 
64  chickens  ;  how  many  fowls  has  he? 

3.  A  man  has  48  cattle  in  one  field  and  36  in  another; 


ADDITION   AND    SUBTRACTION.  17 

if  he  take  15  from  the  second  field  and  put  them  in  the 
first,  how  many  will  then  be  in  each  field  ? 

4.  A  granger  owns  120  cattle,  which  are  pastured  in 
three  fields  ;  in  one  field  there  are  45  cattle,  in  another 
30  cattle ;  how  many  cattle  in  the  third  field  ? 

5.  Willie  has  41  cents'in  one  pocket  and  37  in  another; 
if  he  buy  a  knife  for  50  cents,  how  many  cents  will  he 
have  left  ? 

6.  Willie  had  given  him  two  books  to  read,  one  of 
153  pages  and  the  other  of  226  pages  ;  he  has  read 
240  pages,  how  many  more  has  he  to  read  ? 

7.  In  one  basket  there  are  51  eggs,  in  another  62,  and 
in  a  third  42  eggs ;  how  many  in  the  three  baskets  ? 

8;  One  hen  has  15  chickens,  another  17,  and  a  third 
has  14 ;  how  many  chickens  in  all  ? 

9.  Andrew  learned  75  verses  of  poetry  in  one  week, 
94  in  another,  and  87  in  a  third ;  how  many  verses  did 
he  learn  in  three  weeks  ? 

10.  John  gave  a  beggar  53  cents,  Willie  gave  him 
72  cents,  and  Andrew  gave  him  65  cents ;  how  many 
cents  did  he  receive  ? 

11.  A  man  bought  a  horse  for  95  dollars,  a  cow  for 
40  dollars,  and  a  wagon  for  65  dollars ;  how  much  did 
he  invest  ? 

12.  A  merchant  commenced  business  with  $9,875  ;  the 
first  year  his  net  profits  were  $2,134,  the  second  year 
$1,654,  the  third  year  $2,547,  and  the  fourth  year  $2,620  ; 
what  was  then  the  amount  of  his  funds  ? 

13.  A  merchant  commenced  with  $12,650;  the  first 
year  he  gained  $2,163,  the  second  $1,875,  the  third 
$1,260,  and  the  4th  year  he  lost  $4,105  ;  what  funds  had 
he  left  ? 


18  ADDITION   AND    SUBTRACTION, 

14  A  man  commenced  with  $4500 ;  the  first  year  he 
doubled  his  money,  but  at  the  beginning  of  the  next 
year  he  lost  $2500 ;  he  then  doubled  what  remained,  and 
lost  $6000  ;  then  he  doubled  what  remained  and  closed 
business  with  what  amount  of  money  ? 

15.  A  man  bought  3000  acres  of  land ;  he  then  sold 
to  one  man  324  acres,  to  anather  236  acres,  to  a  third 
148  acres,  to  a  fourth  465  acres,  and  to  a  fifth  634  acres ; 
how  many  acres  were  left  ? 

16.  A  merchant  took  with  him  $5000  to  buy  goods ; 
he  purchased  dry  goods  for  $1864,  groceries  for  $1256 ; 
hardware  for  $630,  hats  and  boots  for  $362  ;  how  much 
had  he  left  ? 

17.  A  man  owns  three  farms ;  the  first  contains 
275  acres,  the  second  483  acres,  and  the  third  1230 
acres;  he  sells  the  first  farm,  236  acres  of  the  second, 
and  584  acres  of  the  third ;  how  many  acres  remain 
unsold  ?  how  many  of  the  second  farm  ?  and  how  many 
of  the  third  ? 

18.  A  man  bought  a  horse  for  $150,  and  a  buggy  and 
harness  for  $275;  he  sold  them,  gaining  $42;  what  did 
he  sell  them  for  ? 

19.  John  Jones  bought  a  farm  for  $6875  ;  he  paid  for 
repairs  $2172,  and  then  sold  it  for  $10165 ;  how  much 
did  he  gain  or  lose  by  the  transaction? 

20.  James  Johnson  bought  one  house  for  $6,540, 
another  for  $7,965,  and  a  third  for  $12,384;  he  paid 
for  repairs  $3,165,  and  then  sold  the  three  houses 
for  $31,236;  how  much  did  he  gain  or  lose  by  the 
transaction  ? 

21.  Invested  3245  dollars  in  property,  and  sold  it  so  as 
to  gain  534  dollars.    What  was  it  sold  for  ? 


ABmTlON  AND  SUB TB ACTION.  19 

2^^.  A  merchant  bought  dry  goods  for  576  dollars,  gro- 
ceries for  $375,  and  hardware  for  $234 ;  in  selling  he  gains 
$156 ;  what  was  the  amount  of  sales  ?       Ans,  $1,341. 

23.  A,  B,  and  C  engage  in  trade;  A  puts  in  $2,576, 
B  $3,845,  and  0  $431  more  than  A  and  B ;  what  is  the 
amount  of  their  stock  ?  Ans.  $13,273. 

24.  A  man  is  9  years  older  than  his  wife,  and  she  is  22 
years  older  than  their  son,  who  is  12  years  old ;  what  is 
the  age  of  each  ?     Ans.  Wife's  age,  34  y.;  man's,  43  y. 

25.  A  man  bequeathed  to  his  wife  $5,650,  to  each  of 
his  three  sons  $3,775,  and  to  each  of  his  two  daughters 
$3,550 ;  what  was  the  amount  of  his  bequests  ? 

Ans.  24,075. 

26.  A  man  owing  $9,856,  pays  at  one  time  $3,453,  and 
at  another  $2,176;  how  much  does  he  still  owe? 

Arts.  $4,227. 

27.  The  sum  of  two  numbers  is  8,643,  and  one  of  the 
numbers  is  5,756;  what  is  the  other  number  ? 

28.  The  number  of  pupils  registered  is  1,253,  and  the 
number  at  school  is  1,168 ;  how  many  are  absent? 

29.  The  sum  of  two  numbers  is  6,432,  and  one  of  the 
numbers  is  2,541 ;  what  is  the  difference  of  the  numbers  ? 

Ans,  1,350. 

30.  The  sum  of  two  numbers  is  5,680,  and  their  differ- 
ence is  596 ;  what  are  the  numbers  ? 

Ans,  3,138  and  2,542. 

31.  The  difference  of  two  numbers  is  265,  and  the 
smaller  number  is  576 ;  what  is  the  larger  number? 

32.  The  difference  of  two  numbers  is  175,  and  the 
larger  number  is  651 ;  what  is  the  sum  of  the  two  num- 
bers? Ans.  1,127. 


20  AVDITION  AND  SUBTRACTIONS 

33.  Add  32,745,  276,  304,721,  5,640,'  216,  243,  976,- 
874,  2,176,  81,275,  9,406,  20,045,  6,320,754,  and  7,105,006. 

Arts,  14,859,377. 

34.  Fifteen  gallons  leaked  out  of  a  cask  of  81  gallons ; 
how  many  gallons  were  left  ? 

35.  A  merchant  engaged  in  trade  with  $5,481,  and 
gained  each  year  for  five  years  $1,254 ;  but  the  sixth  year 
he  lost  $2,162 ;  what  was  then  the  amount  of  his  capital  ? 

Ans.  $9,589. 

36.  A  father  left  an  estate  of  $32,600  to  his  five  chil- 
dren, the  eldest  to  have  $1,000  more  than  the  next,  and 
the  next  $1,000  more  than  his  next  younger  brother,  and 
so  on  to  the  youngest ;  what  was  the  share  of  each  ? 

Ans,  Eldest,  $8,520,  7,520,  6,520,  5,520,  4,520. 

37.  A  man  purchased  1,200  acres  of  land,  and  sold  it  in 
parcels  as  follows  :  to  one  man,  364  acres;  to  another, 
204  acres;  to  a  third,  468  acres,  and  the  balance  (how 
many  acres  ?)  to  a  fourth.  Ans,  164  acres. 

38.  A  man  bought  one  farm  for  $6,748,  and  another 
for  $4,482 ;  he  then  sold  both  for  $12,000 ;  how  much  did 
he  gain  or  lose  ?  Ans,  $770  gain. 

39.  A  man  sold  pork  for  $75;  beef  for  $36;  butter  for 
$25,  and  cheese  for  $18 ;  he  then  bought  sugar  for  $12 ; 
coffee  for  $15 ;  salt  for  $5,  and  dry  goods  for  $31 ;  how 
much  cash  had  he  left?  Ans.  $91. 

40.  Columbus  discovered  America  in  1492,  and  Wash- 
ington achieved  our  Independence  in  1783;  what  time 
elapsed  between  those  two  great  events? 

41.  Washington  died  in  1799,  at  the.  age  of  67  years ;  in 
what  year  was  he  born  ? 

42.  A  merchant  purchased  goods  for  $2,154,  on  which 
he  paid  $1,465  ;  how  much  did  he  still  owe  ? 


MULTIPLICATIOJf. 


Multiplication  and  Division  Table. 


1 

3  3  4|  5  6  7  8 

9 

10 

11 

13 

3 

4  6  8  1 10  1 12  14  16 

18 

20 

22 

24 

3 
4 
5 

6  9  12  15 1 18  21  24 

27 

30 

33 

36 

8  12  16  1  30  1  24  28  32 

36 

40 

44 

48 

10  15  20  1  25  1  30  35  40 

45 

50 

55 

60 

6 

12  18  24  30  36  42  48 

54 

60 

66 

72 

7 

14  21  28 1 35  42  49  56 

63 

70 

77 

84 

8 

16  24  32|40  48  56  64 

72 

80 

88 

96 

9 

18  27  36|45|54|63  72 

81 

90 

99 

108 

lO 

20  30  40  1  50  1  60  70  80 

90 

100 

110 

120 

11 

22  33  44|55  66  77  88 

99 

110 

121 

132 

13 

24  36  48|60  72 1 84  96 

108 

120 

132 

144 

As  a  Multiplication  Table,  begin  with  the  first  line ; 
thus, 

Once  1  is  1 ;  twice  1  are  2 ;  three  times  1  are  3,  etc. 
Second  line,  Once  2  are  2;  twice  2  are  4;  3  tiipoies  2 
are  6 ;  4  times  2  are  8,  etc.  Third  line,  Once  3  are  3 ; 
twice  3  are  6 ;  3  times  3  are  9 ;  4  times  3  are  12,  etc. 
Eecite  each  line  similarly. 

Kem.  4  times  3  are  12,  and  3  times  4  are  12 ;  hence,  alternating 
the  factors  does  not  change  the  product. 


22  MULTIPLICATION, 

As  a  Division  Table,  begin  with  the  first  line ;  thus, 
1  is  contained  in  1,  once;  in  2,  twice;  in  3,  3  times; 
in  4,  4  times,  etc.  Second  line,  2  into  2  =  1 ;  2  into 
4  =  •^;  2  into  6  =  3;  2  into  8  =  4,  etc.  Third  line, 
3  into  3  =  1 ;  3  into  6  =  2,  etc. 

Rp'\i.— As  a  Multiplication  Table,  it  may  also  be  read  by  the 
column,  \>Y  which  the  factors  are  alternated,  without  changing  the 
prod"\ct.  Any  number  is  multiplied  by  10  by  adding  a  zero  to  it. 
As  J'  Division  Table,  the  first  column  has  all  the  divisors,  the  first 
\mc  all  the  quotients,  and  every  number  in  each  line  is  a  dividend, 
wh>c>\  is  always  in  the  same  line  and  the  same  column  with  the 
quotimt  and  divisor.  Any  number  having  a  zero  in  the  units 
placr  is  divided  by  10  by  removing  the  zero. 


THEOREM     I. 

At^y  number  is  jnultipUed  by  10  by  annexing  a 
zere  to  it, 

f^ince  the  product  of  any  number  multiplied  by  1  is 
equal  to  the  number  itself,  the  product  of  any  number 
multiplied  by  2  is  double  the  number,  etc, 

For,  as 
10  X  1  =  10,  and  10  x  2  =  20,  and  10  x  24  =  240, 
and  as  alternating  the  factors  does  not  change  the  product, 
hence, 
1  X  10  =  10,  and  2  x  10  =  20,  and  24  x  10  =  240. 

.-.  Any  number  is  multiplied  by  10  by  annexing  a 
zero  to  it. 

CoK. — Any  number  is  multiplied  by  100  by  annexing 
two  zeros  to  it,  and  annexing  three  zeros  multiplies  it  by 
1000,  etc. 


MULTIPLICATION,  23 


THEOREM     II. 

The  product  of  any  tivo  factors  will  have  as  many 
figures,  or  one  less,  than  both  factors, 

600         500 
5  50 


1 

3 

3 

4 

9 

50 

1 

3 

4 

4 

9 

5 

i 

9 

12 

16 

81 

250 

2500       25000 

The  products  of  the  smaller  figures  of  units  will  be  but 
one  figure  until  above  3,  when  there  will  be  two  figures, 
but  never  more,  as  9  x  9  =  81,  and  every  additional 
figure  annexed  to  each  or  either  factor,  whether  small  or 
large,  will  make  an  increase  of  one  figure  and  no  more ; 
therefore  the  product  of  any  two  factors  will  have  as 
many  figures,  or  one  less  than  both  factors. 

CoE.  1. — The  product  of  any  two  figures  cannot  be  less 
than  one  figure,  nor  more  than  two. 

CoR.  2. — The  product  of  units  by  units  must  be  units, 
and  when  there  are  two  figures,  the  left-hand  figure  will 
be  tens.  The  product  of  tens  by  units  must  be  tens,  and 
when  there  are  two  figures,  the  left-hand  figure  will  be 
hundreds ;  and  if  any  order  be  multiplied  by  units,  the 
right-hand  figure  of  the  product  will  be  the  same  order 
as  the  multiplicand,  and  if  there  be  two  figures  in  the 
product,  the  left-hand  figure  will  belong  to  the  next 
highej  order. 

CoR.  3. — When  the  multiplier  is  tens,  the  product  will 
be  ten  times  as  great  as  if  the  multiplier  were  units ; 
that  is,  each  product  will  have  one  zero  to  the  right  of  it, 
holding  the  units  place,  or  the  first  figure  of  the  product 
must  be  placed  in  the  column  of  tens ;  when  the  multi- 


24  MULTIPLICATION. 

plier  is  hundreds,  the  right-hand  figure  must  be  placed 
in  the  column  of  hundreds;  and,  in  general,  whatever 
the  order  of  the  multiplier  is,  the  right-hand  figure  must 
be  in  the  column  of  that  order. 

CoE.  4. — If  there  be  one  or  more  zeros  in  the  multi- 
jilier,  the  product  of  the  next  figure  will  be  put  back  one 
figure  for  every  zero. 

Rem. — 111  the  multiplication,  each  figure  may  be  regarded  as  the 
unit  of  its  order. 

PROBLEMS. 

1.  10   X   10   =   100. 

2.  11  X  11  =  121  =  11  X  (10  +  1)  =  11  X   1  =    11 

11x10  —  110 

121 

3.  12  X  12  =  144  =  12  X  (10  +  2)  =  12  X    2  z=    24 

12  X 10  =  120 

144 

4.  Multiply  432  by  4  =  (400  +  30  +  2)  x  4. 

2x4=        8         and         432 

30  x  4  =    120  4 

400  X  4  =  1600         1728 
1728 

5.  Multiply  432  by  14  =  432  x  (10  +  4). 

.-.  432  X  4  =  1728     or     432 
432  X  10  =  4320  14 

6048  1728 

432 

6048 
Rem. — The  problems  should  be  carefully  impressed  on  the 
mind  before  proceeding. 


MULTIPLICATION.  25 


6. 


432 

432  X   4  =  1728 

124 

432  X  20  =  8640 

1728 

432  X  100  =  43200 

864 

53568 

432 

53568 

Cor.  1. — When  the  multiplicand  has  several  figures 
and  the  multiplier  one  that  is  only  units,  the  first  product 
of  units  by  units  will  be  units,  or  units  and  tens ;  the 
units  must  be  placed  in  the  right-hand  or  units  place ;  if 
there  be  tens,  it  must  be  reserved  and  placed  in  or  added 
to  the  column  of  tens ;  in  the  next  product  of  tens  by 
units,  the  right-hand  figure  will  be  tens,  and  must  be 
united  with  the  tens  reserved,  and  placed  in  the  column 
of  tens ;  the  left-hand  figure,  if  there  be  one,  must  be 
treated  as  the  previous  one,  reserved  until  the  next 
product  is  obtained,  and  united  with  the  right-hand 
figure;  the  process  is  the  same  in  every  successive 
order. 

Cor.  2. — When  the  multiplier  also  has  several  figures, 
the  process  of  each  successive  multiplier  is  the  same, 
except  that  the  right-hand  figure  of  each  product  must 
be  placed  in  the  order  of  its  multiplier.  (Cor.  3, 
Prob.  2,  page  22.) 

Rem. — A  multiplicand  may  be  eitlier  an  abstract  or  a  concrete 
number,  but  a  multiplier  cannot  be  concrete,  as  it  cannot  refer  to 
tilings,  but  merely  indicates  how  many  times  the  multiplicand  is 
to  be  taken  ;  but  the  product  will  be  of  the  same  name  as  the  mul- 
tiplicand; for  twice  $5  are  $10;  3  times  20  yards  of  cloth  are 
60  yards  of  cloth  ;  twice  4  are  8  ;  3  times  4  are  12,  etc. 

In  computation,  it  is  best  to  regard  all  numbers  as  abstract. 


26 


MVLTTPLICA  TION. 

(7.) 

(8.) 

(9.) 

36435 

26432 

26433 

334 

104 

3004 

145700 

105728 

105728 

72850 

26432 

79296 

109275 

2748928 

79401728 

11801700 

(11.) 

(12.) 

(10.) 

234 

123 

26432 

123 

234 

50004 

702 

492 

105728 

468 

369 

132160 

234 

246 

1321705728 

28783 

28782 

Rem. — The  product  is  not  changed  by  altemathig  the  multipli- 
cand and  multiplier. 


EXAMPLES. 


1.  Multiply   54326  by  346. 

2.  Multiply   23748  by   543. 

3.  Multiply   46874  by   697. 

4.  Multiply   36975  by   476. 

5.  Multiply  236874  by  2134. 


6.  Multiply  9876325  by  35a 

7.  Multiply  879654  by  2175. 

8.  Multiply   986432  by  8704. 

9.  Multiply  326875  by  3005. 
10.  Multiply  468753  by  2100. 


Examples  may  be  added,  or  the  same  repeated,  as  the 
student  will  more  readily  comprehend  by  repetition  than 
by  different  examples. 

Rem.  1. — In  multiplication,  two  factors  are  given  to  find  their 
product. 

Rem.  2. — In  division,  two  numbers  also  are  given  to  find  the 
third ;  the  one  called  the  dividend  corresponds  to  the  product  in 
multiplication,  the  other  given  number  is  called  the  divisor,  and 
the  required  number  is  called  the  quotient ;  the  two  latter  corre- 
spond to  the  factors  in  multiplication. 


Bivisioj^. 


PROBLEMS. 

When  the  product  of  two  numbers  is  4,  and  one  of  the 
numbers  is  2,  the  other  number  is  also  2 ;  for  2x2=4^ 
and  4  divided  by  2,  or  4  divided  into  2  equal  parts,  each 
part  is  2,  that  is,  the  quotient  is  2. 

1.     9  ~  3  =  3.  4.     16  -r-  4  =  4. 


3. 

12  -=-  2  =  6. 

5. 

15  -=-  3  =  5. 

3. 

12  -7-  3  =  4. 

6. 

15  -^  5  =  3. 

CoR.  1. — The  product  of  the  divisor  and  quotient 
equals  the  dividend. 

CoR.    2. — The  divisor  and    quotient   may  be  alter- 
nated. 

24  CoR.  3. — Division  is  the  reverse  of  multi- 

6        plication  and  addition,  and  is  similar  to  sub- 
13        traction  ;  for,  it  is  separating  a  number  in tr 
0        equal  parts,  which  is  the  same  as  subtracting 
rr        the  same  number  from  a  larger  one ;  that  is, 
g        subtracting  the  divisor  from  the  dividend  and 
—        then  from  the  remainder,  repeating  this  pro- 
cess until  there  is  no  remainder,  or  until  the 
remainder  is  less  than  the  divisor.     6  is  sub- 
^        tracted  4  times,  hence  it  is  contained  four 
times.    24  ~  6  =  4. 


28 


DIVISION. 

(1-) 

(2.) 

(3.) 

10 )  100  ( 10 

11 )  121 

(11 

12  )  144  ( 12 

10 

11 

12 

0 

11 

24 

11 

24 

(4.) 

(5.) 

11 )  121  (  10  +  1 

12  )  144  (  10  +  3 

110 

120 

11 

24 

11 

24 

6,  48  ->  13  =  4. 

12. 

120  -^  10  =  12. 

7.  64  -J-  8  =  8. 

13. 

130  -^  10  =  13. 

8.  96  H-  12  =  8. 

14. 

140  -f-  10  =  14. 

9,  12  X  4  =  48. 

15. 

10  X  13  =  120. 

0.   8x8  =  64. 

16. 

10  X  13  =  130. 

.1.  12  X  8  =  96. 

17. 

10  X  14  =  140. 

Cor.  1. — Adding  a  zero  to  the  right  of  a  number  mul- 
tiplies the  number  by  10 ;  taking  a*  zero  away  from  the 
right  of  a  number  divides  the  number  by  10. 
Divide  60536  by  4;  thus, 

or    4  )  60536 
15134 


)  60536  { 

10000 

40000 

30536  ( 

5000 

30000 

536  ( 

100 

400 

136  ( 

30 

120 

16  ( 

4 

16 

15134 

The  divisor  4  is  contained  once 
in  the  unit  of  the  highest  order  of 
the  dividend,  which  is  one  ten- 
thousand  ;  into  the  remainder 
5000  times,  then  100,  30  and 
lastly  4. 


DIVISION,  29 

Rem.  1. — The  same  result  is  obtained  by  short  division,  by 
putting  the  first  figure  of  the  quotient  under  the  left-hand  figure 
of  the  dividend  (when  it  is  contained  in  it),  as  it  is  of  the  same 
order. 

Rem.  8. — If  the  unit  of  the  divisor  is  not  contained  in  the  first 
unit  of  the  dividend,  then  the  first  figure  of  the  quotient  will  be 
of  the  same  order  as  the  second  figure  of  the  dividend  and  should 
be  placed  under  it. 


ivide  60536  by  14 ;  thus, 

14  )  60536  (  4324   and 

214  )  925336  (  4324 

56 

856 

45 

693 

43 

643 

33 

513 

28 

428 

56 

856 

56 

856 

4334  X  14  =  60536. 

4334  X  214  =  925336. 

Cor.  1. — Since  the  product  of  any  two  factors  will 
have  as  many  figures  or  one  less  than  both  factors,  so  in 
division  the  number  of  figures  of  the  divisor  and  quotient 
will  either  be  equal  to  or  one  greater  than  that  of  the 
dividend. 

CoR.  2. — When  the  divisor  is  contained  in  the  same 
number  of  figures  of  the  dividend  as  is  in  the  divisor,  then 
the  number  of  figures  of  the  divisor  and  quotient  will  be 
one  more  than  that  of  the  dividend  ;  but  when  it  requires 
an  additional  figure  of  the  dividend  to  contain  the 
divisor,  then  the  number  of  figures  of  fche  divisor  and 
quotient  will  be  equal  to  that  of  the  dividend. 


30  DIVISION. 

PROBLEMS. 

1.  Divide  9253360  by  2140;  thus,  (2.) 

21410  )  925336|0  ( 4324  26432 

856  '  104 

105728 
26432 

26432)2748928(104 
26432 

105728 
105728 

3.  Divide  987654321  by  12300. 

123100  )  9876543|21  (  80297 '  80297 

984  12300 

365  240891 

246  160594 

1194  80297 


693 

4324 

642 

2140 

513  ^ 

17296 

428 

4324 

856 

8648 

856 

9253360 

1107  987653100 

873  1221 

861  987654321 

1221,  remainder. 

Rem.  1. — When  the  dividend  is  not  the  exact  product  of  two 
integral  numhers,  there  will  be  a  remainder,  and  the  dividend  is 
equal  to  the  product  of  the  quotient  and  divisor  plus  the 
remainder. 

Rem.  2. — When  there  are  the  same  number  of  zeros  in  divi- 
dend and  divisor,  beginning  with  the  order  of  units  they  may  be 
canceled  ;  and  when  there  are  zeros  in  the  divisor  only,  they  may 
be  omitted,  and  also  the  same  number  of  figures  in  the  dividend, 
which  after  the  division  is  performed  must  be  brought  down  as  a 
part  or  the  whole  of  the  remainder. 


DIVISION.  31 

EXAMPLES. 

1.  Divide       235643  by  123. 

2.  Divide       345678  by  234. 

3.  Divide       234567  by  891. 

4.  Divide      1357916  by  248. 

5.  Divide  369875432  by  1768. 

6.  Divide  487698425  by  625. 

7.  Divide  987654321  by  1234. 

8.  Divide  876543219  by  2345. 

9.  Divide  678956732  by  1546. 

10.  Divide    34567890  by       2564. 

11.  Divide  786954321  by    176543. 

12.  Divide  678900432  by  1004000. 

The  student  must  not  proceed  until  he  is  famihar 
with  division. 

EXAMPLES. 

1.  What  cost  5  lbs.  of  sugar  at  10  cts.  per  lb.  ? 

2.  At  10  cts.  per  lb.,  how  many  lbs.  can  be  bought 
for  50  cts.  ? 

3.  What  cost  10  lbs.  of  sugar  at  10  cts.  per  lb.  ? 

4.  At  10  cts.  per  lb.,  how  many  lbs.  can  be  bought 
for  100  cts.  ? 

5.  What  cost  15  lbs.  of  sugar  at  10  cts.  per  lb.  ? 

6.  At  10  cts.  per  lb.,  how  many  lbs.  can  be  bought 
for  150  cts.  ? 

7.  What  cost  20  lbs.  of  sugar  at  10  cts.  per  lb.  ? 

8.  At  10  cts.  per  lb.,  how  many  lbs.  can  be  bought 
for  200  cts.  ? 

9.  What  cost  25  lbs.  of  sugar  at  10  cts.  per  lb.  ? 

10.  At  10  cts.  per  lb.,  how  many  lbs.  can  be  bought 
for  250  cts.  ? 


32  DIVISION. 

11.  What  cost  245  lbs.  of  beef  at  8  cts.  per  lb.  ? 

12.  At  8  cts.  per  lb.,  how  many  lbs.  of  beef  can  be 
bought  for  1960  cts.  ? 

13.  What  cost  348  acres  of  land  at  $45  per  acre  ? 

14.  At  $48  per  acre,  how  many  acres  can  be  bought 
for  $15660  ? 

15.  What  cost  3245  acres  of  land  at  $64  per  acre  ? 

16.  At  $64  per  acre,  how  many  acres  can  be  bought 
for  $207680  ? 

17.  What  is  the  cost  of  15  horses  at  $125  each  ? 

18.  How  many  horses  at  $125  each  can  be  bought 
for  $1875  ? 

19.  What  is  the  cost  of  35  oxen  at  $75  each  ? 

20.  How  many   oxen   at    $75  each   can   be    bought 
for  $2625  ? 

21.  What  is  the  cost  of  84  cows  at  $45  each  ? 

22.  How   many   cows   at    $45  each   can   be   bought 
for  $3780  ? 

.    23.  Multiply  54682  by  9  ;  thus, 


54682  X  10 

=  646820 

54682  X  1 

=   54682 

492138 

24.  Multiply  54682  by  99. 

54682  X  100 

=  5468200 

54682  X   1 

=    54682 

5413518 

25.  Multiply  54682  by  999. 

54682  X  1000  =     54682000 

54682  X    1  =    54682 

54627318 


DIVISION,  33 

26.  Multiply  54682  by  25.  Instead  of  25,  multiply 
T)y  J-f^.  4  )  5468200 

1367050 

27.  Multiply  54682  by  50.  Instead  of  50,  multiply 
byH^. 

28.  Multiply  54682  by  75.  Multiply  the  result  ot 
26th  by  3. 

29.  Multiply  54682  by  150. 

54682    X   100     =     5468200 
Adding  |,  2734100 

8202300 

30.  Divide  546825  by  25.  Multiply  by  4  and  strike  off 
two  figures. 

31.  Divide  546850  by  50.  5468|50 

2 

10937.00 

Rem. — As  a  number  is  multiplied  by  100  by  adding  two  zeros, 
so  any  number  is  divided  by  100  by  cancelling  the  two  right-hand 
figures,  which  will  then  form  the  remainder. 

32.  Bought  24  horses  at  $84  each,  54  cows  at  $36 
each,  and  364  sheep  at  $4  each ;  what  did  the  horses 
cost  ?  the  cows  ?  the  sheep  ?  what  did  all  cost  ? 

33.  Sold  all  the  stock  of  the  last  example  at  a  profit 
of  $324  ;  what  did  I  sell  them  for  ? 

34.  Bought  3064  acres  of  land  at  $25  per  acre,  and 
sold  it  at  a  loss  of  $1245.  What  did  the  land  cost  ?  and 
what  was  it  sold  for  ? 

35.  Bought  3840  acres  of  land  at  $10  per  acre ; 
divided  it  into  twenty  farms  of  an  equal  number  of 


34  DIVISION, 

acres  each;  twelve  of  the  farms  I  sold  at  $15  per 
acre,  3  of  the  farms  at  ^12  per  acre,  and  the  five 
remaining  farms  at  $4  per  acre;  did  I  gain  or  lose, 
and  how  much  ? 

36.  A  farm  of  364  acres  was  bought  for  $9100  and 
sold  at  a  gain  of  $1456;  what  was  paid  per  acre  ?  and  at 
what  rate  was  it  sold  ? 

37.  Bought  184  acres  of  land  for  $11960,  and  sold  it 
for  $13064;  what  was  the  cost, per  acre?  aod  at  what 
rate  was  it  sold  ? 

38.  The  cost  of  12  horses  and  15  cows  was  $2760 ;  the 
horses  cost  $180  each;  what  was  the  average  cost  of 
each  cow  ? 

39.  In  a  square  mile  there  are  640  acres ;  how  many 
farms  of  160  acres  each  in  a  State  that  has  9,000  square 
miles  ?  how  many  in  one  of  46,000  sq.  miles  ?  how  many 
in  one  of  257,000  sq.  miles  ? 

40.  What  is  the  value  of  the  land  in  the  first  State 
at  $25  per  acre?  in  the  second  at  $20  per  acre  ?  and  in 
the  third  at  $5  per  acre  ? 

41.  A  man  having  an  estate  worth  $15,000,  increases 
it  $2500  every  year  for  twenty-five  years,  when  he  dies, 
leaving  it  as  follows :  To  his  wife  $20,000,  to  his  eldest 
son  $8000,  to  his  second  son  $7000,  to  his  third  son 
$7000,  and  the  remainder  to  be  divided  equally  among 
his  five  daughters ;  what  is  the  share  of  each  daughter  ? 

42.  A  merchant  commences  business  with  a  capital  of 
$25000  ;  at  the  end  of  the  first  year  he  finds  that  he  has 
increased  it  $5000,  the  second  year  the  increase  is  $4500, 
and  the  same  at  the  end  of  the  third  year,  when  he  trans- 
fers the  whole  business  to  his  three  sons.  What  is  the 
capital  of  each  son  ?  Arts,  $13000. 


Review. 


NUMEEATION". 

Write  in  figures  the  following  numbers : 

Twenty-five,  One  hundred  and  twenty-five,  Two  hun- 
dred and  five,  Three  hundred  and  six.  Four  hundred  and 
eight,  One  thousand,  One  thousand  three  hundred  and 
forty-five.  Five  thousand  eight  hundred  and  four.  Ten 
thousand  five  hundred  and  nine.  Thirty-four  thousand 
and  twenty,  Fifty-three  thousand  and  five.  Five  hundred 
thousand,  Six  hundred  thousand  and  four,  One  million, 
One  million  four  hundred  and  twenty-five  thousand  six 
hundred  and  twelve. 

ADDITION 

ORAL     QUESTIONS. 

John  has  8  apples,  WilHe  has  7,  and  Andrew  5 ;  how 

many  apples  have  the  three  boys  ? 

Bought  coal  for  $8,  wood  for  $6,  and  a  barrel  of  flour 

for  $5 ;  what  was  the  amount  of  the  purchase  ? 
How  many  are 

6  +  4  +  5      2  +  5  +  8      8  +  7—5  5  +  6  +  8—5 

8  +  7  +  4      5  +  9  +  2       9  +  8—7  4  +  5  +  9—8 

3+8+7      1+3+6      7+9+8  3+9+7-9 

4  +  6  +  9       9  +  6  +  4       6  +  5  +  4—6     2  +  7  +  8—4 
How  many  are  3  +  7;  13  +  7;  23  +  7;  43  +  7;  53  +  7; 

63  +  7;  73  +  7;  83  +  7;  93  +  7? 
Howmany  are  3+8;  13  +  8;  23  +  8;  33  +  8;  43  +  8; 

53  +  8;  63  +  8;  73  +  8;  83  +  8;  93  +  8? 


36  ADDITION. 

How  many  are  4  +  7;  14  +  7;  24  +  7;  34+7;  44  +  7; 

54  +  7;  64  +  7;  74+7;  84+7;  94  +  7? 

How  many  are  5  +  7;  15  +  7;  25  +  7;  35  +  7;  45  +  7; 

55  +  7;  65  +  7;  75  +  7;  85  +  7';  95  +  7? 

How  many  are  5  +  8;  15  +  8;  25  +  8;  35  +  8;  45  +  8; 
55  +  8;  65  +  8;  75  +  8;  85  +  8;  95  +  8? 

How  many  are  5  +  9;  15  +  9;  25  +  9;  35  +  9;  45  +  9; 

55  +  9;  65  +  9;  75  +  9;  85  +  9;  95  +  9? 

How  many  are  6  +  7;  16  +  7;  26  +  7;  36  +  7;  46  +  7; 

56  +  7;  66  +  7;  76  +  7;  86  +  7;  96+7? 

How  many  are  6  +  8;  16  +  8;  26  +  8;  36  +  8;  46  +  8; 

56  +  8;  66  +  8;  76  +  8;  86  +  8;  96  +  8? 

How  many  are  7  +  7;  17  +  7;  27  +  7;  37  +  7;  47  +  7; 

57  +  7;  67  +  7;  77  +  7;  87  +  7;  97  +  7? 

How  many  are  7  +  8;  17  +  8:  27  +  8;  37  +  8;  47  +  8; 
57  +  8;  67  +  8;  77  +  8;  87  +  8;  97  +  8? 

These  questions  should  be  frequently  repeated  or  simi- 
lar ones  asked. 


EXAMPLES. 

Add  the  foUowinj 

y  numbers: 

1 

2.10 

3.100 

4.111 

2 

20 

200 

223- 

After  adding  the  first 

3 

30 

300 

333 

three  examples,  add  their 

4 

40 

400 

444 

sums,  and  observe  the  cor- 

5 

50 

500 

555 

respondence  with  Ex.  4, 

6 

60 

600 

666 

when  the  tens  and  hun- 

7 

70 

700 

777 

dreds  are  added  with  the 

8 

80 

800 

888 

columns  of  tens  and  hun- 

9 

90 

900 

999 

dreds. 

SUBTRACTION. 

6.  28 

6.  53 

7.  45        8.  56 

9.  79 

36 

U 

61             43 

86 

37 

10.  84 
73 


11.  121    12.  312    13.  276    14.  751     15.  542     15.  374 
343  256  543  324  324  236 

621  461  631  263  261  573 

17.  What  is  the  sum  of  4,321,  604,  and  36  ? 

18.  What  is  the  sum  of  264,  301,  and  5  ? 

19.  What  is  the  sum  of  205,  25,  and  6  ? 

20.  What  is  the  sum  of  4,  24,  324  ? 

21.  What  is  the  sum  of  3,276,  2,132,  and  5,437  ? 

22.  What  is  the  sum  of  57,896,  98,765,  and  78,901  ? 

Similar  examples  emi  larger  ones  may  be  given  for  the 
slate  or  blackboard. 


SUBTRACTION. 

ORAL      QUESTIONS. 

Take  7  from  9;  7  from  19;  7  from  29;  7  from  39;  7 
from  49 ;  7  from  59  ;  7  from  69 ;  7  from  79 ;  7  from  89 ; 

7  from  99  ;  7  from  109. 

Take  6  from  13 ;  6  from  23  ;  6  from  33;  6  from  43 ;  6 
from  53 ;  6  from  63 ;  6  from  73 ;  6  from  83 ;  6  from  93 ; 
6  from  103. 

Take  8  from  14;  8  from  24;  8  from  34;  8  from  44 ;  8 
from  54;  8  from  64;  8  from  74;  8  from  84;  8  from  94; 

8  from  104. 

Take  9  from  16 ;  9  from  26 ;  9  from  36 ;  9  from  46 ;  9 
from  56;  9from  66;  9  from  76  j  9  from  86;  9  from  96; 

9  from  106. 


38 


SUBTRACTION. 


Take  2  from  11 ;  2  from  21 ;  2  from  31 ;  2  from  41 ;  2 
51;  2  from  61;  2  from  71;  2  from  81;  2  from  91;  2 
from  101. 

Take  3  from  12 ;  3  from  22  ;  3  from  32 ;  3  from  42 ;  3 
from  52  ;  3  from  62 ;  3  from  72 ;  3  from  82 ;  3  from  92 ; 

3  from  102. 

Take  4  from  13  ;  4  from  23 ;  4  from  33 ;  4  from  43 ;  4 
from  53 ;  4  from  63 ;  4  from  73  ;  4  from  83  ;  4  from  93 ; 

4  from  103. 

Take  5  from  14 ;  5  from  24 ;  5  from  34 ;  5  from  44 ;  5 
from  54 ;  5  from  64 ;  5  from  74 ;  5  from  84 ;  5  from  94 ; 

5  from  104. 

Take  2  from  10 ;  2  from  20 ;  2  from  30 ;  2  from  40 ;  2 
from  50 ;  2  from  60 ;  2  from  70 ;  2  from  80 ;  2  from  90 ; 
2  from  100. 

These  questions  should  be  ofteu  repeated. 


EXAMPLES. 

1. 

From 

10  take  1 ; 

2,  3,  4,  5,  6,  7,  8,  9. 

2. 

From 

20  take.  1 . 

2,  3,  4,  5,  6,  7,  8,  9. 

3. 

From 

30  take  1 

;  2,  3,  4,  5,  6,  7,  8,  9. 

4. 

From 

40  take  1 ; 

2,  3,  4,  5,  6,  7,  8,  9. 

5. 

From 

50  take  1 

2,  3,  4,  5,  6,  7,  8,  9. 

6. 

From 

60  take  1 , 

2,  3,  4,  5,  6,  7,  8,  9. 

7. 

From 

70  take  1 . 

,  2,  3,  4,  5,  6,  7,  8,  9. 

8. 

From 

80  take  1 , 

2,  3,  4,  5,  6,  7,  8,  9. 

9. 

From 

90  take  1 

,  2,  3,  4,  5,  6,  7,  8,  9. 

10. 

From  100  take  1 ; 

2,  3,  4,  5,  6,  7,  8,  9,  99. 

11. 

From  200  take  1 , 

2,  3,  4,  5,  6,  7,  8,  9,  199. 

12. 

From  300  take  1 

;  2,  3,  4,  6,  6,  7,  8,  9,  299. 

13. 

From  400  take  1 

;  2,  3,  4,  5,  6,  7,  8,  9,  399. 

SUBTRACTION.  39 

14.  From  1,000  take  1 ;  2,  3,  4,  5,  6,  7,  8,  9,  999. 

15.  From  10,000  take  1 ;  2,  3,  4,  5,  6,  7,  8,  9,  9,999. 

16.  From  100,000  take  99,999. 

17.  From  2,000,000  take  1,999,999. 

18.  From  3,246,532  take  2,164,320. 

19.  From  800  take  400. 

20.  From  8,000  take  4,000. 

21.  From  545  take  116. 

22.  From  576  take  144. 

23.  From  624  take  304. 

24.  From  5,324  take  1,304. 

25.  From  6,725  take  2,432. 

26.  From  74,832  take  68,741. 

27.  From  83,754  take  48,623. 

28.  Columbus  discovered  America  in  the  year  1492; 
how  many  years  since  ? 

29.  Washington  was  born  in  1732,  and  lived  67  years; 
in  what  year  did  he  die  ? 

30.  I  owed  $1,000,  and  I  paid  one  man  $324,  another 
$204,  a  third  $120,  and  a  fourth  $67;  how  much  do  I 
still  owe?  Ans.  $285. 

31.  A  man  dying  leaves  $5,000  to  his  widow  and  son, 
the  widow  to  have  $1,000  more  than  the  son ;  what  is  the 
portion  of  each  ?     Ans.  Son,  $2,000 ;  widow,  $3,000. 

32.  A  father  divides  his  estate  of  $12,000  as  follows : 
To  each  of  his  two  daughters,  $2,000  ;  to  one  son,  $1,500 ; 
to  another,  $2,000 ;  to  a  third  son,  $2,500,  and  the  bal- 
ance to  the  widow ;  what  is  her  share  ?     Ans.  $2,000. 


4:0  ADDITION  AND   SUB1*RACTI0N. 

ADDITION  AND  SUBTKACTION. 

EXAMPLES, 

1.  A  man  has  six  farms;  in  the  1st  6,312  acres ;  in  the 
2d  3,241  acres ;  in  the  3d  4,276  acres ;  4th,  272  acres ; 
5th,  304  acres,  and  in  the  6th,  63  acres ;  how  many  acres 
in  all  ?  Ans.  14,468  acres, 

s^  2.  A  merchant  sold  goods  for  $6,032,  gaining  thereby 
$326.     What  did  the  goods  cost  ?  Ans,  $5,706. 

3.  A  man  owning  2,142  acres  of  land,  gave  to  his  eldest 
son  834  acres,  to  his  second  son  612  acres,  and  to  the 
third  420  acres ;  how  many  acres  were  left  ? 

Ans.  276  acres. 
?^4.  A  man  left  real  estate  worth  $9,376,  and  personal 
property  worth  $2,142  ;  he  owes  one  man  $1,236,  another 
$875,  and  a  third  $450 ;    what  is  the  net  value  of  his 
estate?  Ans.  88,957. 

5.  A,  B,  and  C  commenced  business ;  A  put  in  $3,275, 
B  $4,150,  and  C  $5,180;  when  they  closed  they  found 
that  they  had  lost  $1,200;  what  had  they  at  the  begin- 
ning and  at  the  close  ? 

Alls.  Commenced  with  $12,605  and  ended  $11,405. 
^6.  A  horse  and  buggy  together  are  worth  $500;  but 
the  horse  is  worth  $100  more  than  the  buggy ;  what  is 
the  value  of  each  ?      A7is.  Horse,  $300 ;  buggy,  $200. 

7.  The  sum  of  two  numbers  is  5,439,  and  the  one 
number  is  215  greater  than  the  other;  what  are  the 
numbers  ? 

8.  The  minuend  is  5,746,  and  the  subtrahend  3,825 ; 
what  is  the  difference  ? 

y   9.  The  subtrahend  is  3,825,  and  the  difference  1,921 ; 
what  is  the  minueud  ? 


MULTIPLICATION,  41 

10.  The  minuend  is  5,746  and  the  difference  1,921; 
what  is  the  subtrahend  ? 

11.  When  you  know  the  minuend  and  difference,  how 
do  you  find  the  subtrahend  ? 

12.  How  do  you  find  the  minuend,  knowing  the  sub- 
trahend and  difference  ? 

MULTIPLICATIOK 

ORAL       EXERCISES. 

I 

If  an  apple  is  worth  2  cents,  what  are  2  apples  worth  ? 
3?4?5?6?7?8?9?10?11?12? 

If  an  apple  is  worth  3  cents,  what  are  2  apples  worth  ? 
3?  4?  5?  6?  7?  8?  9?  10?  11?  12? 

If  an  apple  is  worth  4  cents,  what  are  2  apples  worth  ? 
3  ?  4  ?  5  ?  6  ?  7  ?  8  ?  9  ?  10  ?  11  ?  12  ? 

If  an  apple  is  worth  5  cents,  what  are  2  apples  worth  ? 
3  ?  4  ?  5  ?  6  ?  7  ?  8  ?  9  ?  10  ?  11  ?  12  ? 

If  an  apple  is  worth  6  cents,  what  are  2  apples  worth  ? 
3?  4?  5?  6?  7?  8?  9?  10?  11?  12? 

If  an  apple  is  worth  7  cents,  what  are  2  apples  worth  ? 
3  ?  4  ?  5  ?  6  ?  7  ?  8  ?  9  ?  10  ?  11  ?  12  ? 

If  an  apple  is  woiTth  8  cents,  what  are  2  apples  worth  ? 
^3?  4?  5?  6?  7?  3?  9?  10?  11  ?  12? 

If  an  apple  is  worth  9  cents,  what  are  2  apples  worth  ? 
3?  4.^  5?  6?  7?  8  ?  9?  10?  11  ?  12? 

If  an  apple  is  worth  10  cents,  what  are  2  apples  worth  ? 
3?  4?  5?  6?  7?  8?  9?  10?  11?  12  ? 

If  an  apple  is  worth  11  cents,  what  are  2  apples  worth  ? 
3?  4?  5  ?  6?  7?  8?  9?  10?  11?  12? 

If  an  apple  is  worth  12  cents,  what  are  2  apples  worth  ? 
3  ?  4  ?  5  ?  6  ?  7  ?  8  ?  9  ?  10  ?  11  ?  12  ? 


48  MULTIPLICATION, 

EXAMPLES. 

1.  Multiply  13  by  2,  3,  4,  5,  6,  7,  8,  9,  10, 11, 12. 

2.  Multiply  14  by  2,  3,  4,  5,  6,  7,  8,  9,  10,  11, 12. 

3.  Multiply  15  by  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12. 

4.  Multiply  16  by  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12. 

5.  Multiply  17  by  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12. 

6.  Multiply  18  by  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12. 

7.  Multiply  256  by  4,  24,  124. 

8.  Multiply  2,145  by  4,  24, 124. 

9.  Multiply  3,636  by  5,  20,  100,  125. 

10.  Multiply  6,435  by  5,  20,  300,  1,000,  and  by  1,325„ 
Analyze. 

11.  Multiply  3,674  by  104,  206,  3,005. 

Ans,  382,096,  756,844,  and  11,040,370. 

12.  Multiply  21,564  by  2,100,  2,010,  2,001. 

Ans.  45,284,400,  43,343,640,  and  43,149,564. 

13.  Multiply  54,000  by  1,200,  14,000,  630. 

Ans.  64,800,000,  756,000,000,  and  34,020,000. 

14.  Multiply  60,401  by  1,040,  1,004,  1,400. 

Ans.  62,817,040,  60,642,604,  and  84,561,400. 

15.  Multiply  four  thousand  and  three  by  three  hundred 
and  one.  •    Ans,  1,204,903. 

16.  Multiply  forty-six  thousand  one  hundred  and 
three  by  three  thousand  and  two.       A71S,  138,401,206. 

17.  Multiply  four  hundred  and  fifty-six  thousand  three 
hundred  and  fifty-four  by  three  thousand  five  hundred 
and  thirty-six.  Ans.  1,613,667,744. 

18.  Multiply  four  millions  three  hundred  and  fifty-four 
thousand  six  hundred  and  twenty-five  by  thirty-six  thou- 
sand four  hundred  and  sixty-seven. 

Ans.  158,800,109,875. 


DIVISION.  43 

19.  Multiply  sixty-four  millions  seven  hundred  and 
ninety-five  thousand  eight  hundred  and  sixty-nine  by 
six  hundred  and  fifty-four  thousand  three  hundred  and 
seventy-eight.  Ans.  42,400,991,164,482. 

20.  Multiply  six  hundred  and  four  millions  two  hun- 
dred thousand  four  hundred  and  four  by  four  hundred 
and  five  thousand  two  hundred. 

Ans.  244,822,003,700,800. 

21.  What  is  the  product  of  3,754  and  268. 

A71S,  1,006,072. 

22.  The  factors  of  a  number  are  57  and  29 ;  what  is  the 
number?  Ans,  1,653. 

23.  Two  of  the  three  factors  of  the  number  1,188  are  9 
and  12 ;  what  is  the  third  factor  ?  Ans.  11. 

DIVISION. 

ORAL     EXERCISES. 

If  one  apple  cost  2  cents,  how  many  .can  you  buy  for 
4  cents?  6?  8?  10?  12? 

How  often  is  2  contained  in  4  ?  6  ?  8  ?  10  ?  12  ?  14  ? 
16  ?  18  ?  20  ?  22  ?  24  ? 

How  often  is  3  contained  in  6  ?  9  ?  12  ?  15  ?  18  ?  21  ? 
24?  27?  30?  33?  36? 

How  often  is  4  contained  in  8  ?  12  ?  16  ?  20  ?  24  ?  28  ? 
32?  36?  40?  44?  48? 

How  often  is  5  contained  in  10?  15?  20  ?  25  ?  30? 
35  ?  40  ?  45  ?  50  ?  55  ?  60  ? 

How  often  is  6  contained  in  12  ?  18  ?  24  ?  30  ?  36  ?  42  ? 

48  ?  54  ?  60  ?  66  ?  72  ? 

How  often  is  7  contained  in  14  ?  21  ?  28?  35  ?  42  ? 

49  ?  56  ?  63  ?  70  ?  77  ?  84  ? 


44  MULTIPLICATION  AND   DIVISION. 

How  often  is  8  contained  in  16  ?  24  ?  32  ?  40  ?  48  ? 
56  ?  64  ?  72  ?  80  ?  88  ?  96  ? 

How  often  is  9  contained  in  18  ?  27  ?  36  ?  45  ?  54  ? 
63  ?  72  ?  81  ?  90  ?  99  ?  108  ? 

How  often  is  10  contained  in  20  ?  30  ?  40?  50?  60? 
70?  80?  90?  100?  110?  120? 

How  often  is  11  contained  in  22?  33  ?  44?  55?  66? 
77?  88?  99?  110?  121?  132? 

How  often  is  12  contained  in  24?  36?  48?  60?  72? 
84?  96?  108?  120?  132?  144? 

Divide  12  into  2  equal  parts;  into  3,  4,  6. 

Divide  24  into  2  equal  parts ;  into  3,  4,  6. 

Divide  36  into  2  equal  parts;  into  3,  4,  6. 

Divide  48  into  2  equal  parts ;  into  3,  4,  6-. 

If  10  yards  of  cloth  cost  $50,  what  is  the  cost  of  one 
yard? 

If  4  pecks  make  one  bushel,  how  many  bushels  in  36 
pecks  ? 

If  48  bushels  of  wheat  cost  $48,  what  is  the  cost  of  one 
bushel  ? 

If  24  bushels  of  wheat  cost  $48,  what  is  the  cost  of  one 
bushel  ? 

Divide  48  apples  equally  among  4  boys ;  6  boys ;  8,  12. 

MULTIPLICATIOlSr  AND  DIVISION. 

EXAM  PLE  S. 

1.  At  $5  each,  how  many  sheep  can  be  bought  for 
$25?  $50?  $100?  $200?  $300?  $400?  $500?  $600?  $625? 
$650?  $675? 

2.  At  $6  each,  how  many  for  $24  ?  $48?  $96?  $120? 
$132  ?  $144  ?  $240  ?  $264  ?  $300  ?  $336  ?  $348  ? 


MULTIPLICATION  AND  DIVISION,  45 

3.  Divide  $1,200  into  5  equal  parts;  how  many  in  each 
part  ? 

4.  Divide  48,000  into  4  parts,  or  two  equal  pairs,  so 
that  each  number  of  the  first  pair  shall  be  one-half  of  one 
of  the  numbers  of  the  second  pair. 

Ans,  Each  of  the  first  pair  8,000,  and  each  of  the  sec- 
ond 16,000. 

5.  Divide  24,000  into  three  parts,  so  that  the  difference 
of  any  two  consecutive  parts  shall  be  1,000. 

Ans.  7,000,  8,000,  and  9,000. 

6.  How  long  would  it  take  a  man  to  travel  around  the 
earth,  whose  circumference  is  about  25,000  miles,  if  he 
travel  50  miles  a  day  ? 

7.  A  man  purchased  the  same  number  of  horses  and 
cows  for  $5,640,  the  horses  at  $85  each,  and  the  cows  at 
$35  each.    How  many  of  each  kind  ?   Ans,  47  of  each. 

8.  A  drover  bought  sheep,  hogs,  and  cows,  of  each  an 
equal  number,  for  $2,752,  the  sheep  at  $4  each,  the  hogs 
at  $11,  and  the  cows  at  $28  each;  how  many  of  each 
kind  ?  Ayis,  64  of  each. 

9.  The  divisor  is  463  and  the  quotient  264;  what  is 
the  dividend?  Ans.  122,232. 

10.  The  quotient  is  327  the  dividend  160,557  ;  what  is 
the  divisor  ?  Aiis.  491. 

11.  A  farmer  sold  8  cows  and  12  horses  for  $1,092 ;  the 
horses  were  rated  at  $75  each ;  what  were  the  cows  sold 
at?  Ans.  $24. 

12.  A  man  bought  a  farm  of  225  acres  for  $10,125,  and 
sold  it  for  $11,250 ;  what  did  he  pay  per  acre,  and  how 
much  per  acre  did  he  gain  ? 

Ans.  He  paid  $45  and  gained  $5  per  acre. 


46  MULTIPLICATION  AND  DIVISION. 

13.  A  clerk  has  a  salary  of  $80  per  month ;  he  pays  for 
board  $25  per  month  and  other  expenses  $15  per  month; 
how  much  does  he  save  per  month  ?  how  much  per  year  ? 

Ans,  $40  per  month  and  $480  per  year. 

14.  If  I  pay  $225  for  75  sheep,  and  they  cost  $1  each  for 
keeping,  and  I  sell  them  at  $5  each,  how  much  is  my 
profit  ?  Ans.  $75. 

15.  If  8  cords  of  wood  cost  $32,  what  is  the  cost  of  1 
cord?  of  2  cords?  3  ?  4?  5?  6?  7  ?  8?  9  ? 

16.  If  9  lbs.  sugar  cost  63  cents,  what  is  the  cost  of  11 
lbs.? 

17.  If  12  lbs.  coffee  cost  300  cents,  what  cost  5  lbs.? 

18.  If  8  lbs.  wool  cost  280  cents,  what  cost  15  lbs.  ? 

Ans.  525  cents. 

19.  A  farmer  sold  235  barrels  of  flour  at  $6  per  barrel ;  he 
bought  goods  for  $830,  and  for  the  balance  took  coal  at  $4 
per  ton  ;  how  many  tons  did  he  get  ?    Ans,  145  tons. 

20.  The  quotient  is  164,  the  divisor  235,  and  the  re- 
mainder 143;  what  is  the  dividend?  Ans,  38,683. 

21 .  How  many  revolutions  will  a  wheel  20  feet  in  cir- 
cumference make  in  running  a  mile  (5,280  ft.)  ? 

Ans,  264. 

22.  If  two  men  can  do  a  piece  of  work  in  3  days,  how 
long  will  it  take  one  man  to  do  it  ?  Ans.  6  days. 

23.  If  4  men  can  do  a  piece  of  work  in  6  days,  how  long 
will  it  take  3  men  to  do  it?  Ans,  8  days. 

24.  If  15  acres  of  land  cost  $375,  what  is  the  cost  of  1 
acre  ?  2  acres  ?  3  acres  ?  4  acres  ?  5  acres  ?  55  acres  ? 

Ans,  $25,  $50,  $75,  $100,  $125,  $1,375. 

25.  If  5  tons  of  hay  cost  $60,  what  is  the  cost  of  1  ton  ? 
2  tons?  3  tons?  45  tons?  75  tons? 

Ans,  $12,  $24,  $36,  $540,  $900. 


MULTIPLICATION  AND  DIVISION.  47 

26.  If  36  acres  of  land  cost  $1,440,  what  will  57  acres 
cost?  84  acres?  Arts,  $2,280  and  $3,360. 

27.  Sold  12  horses  at  $75  each  and  15  cows  at  $24  each, 
and  invested  the  proceeds  in  sheep  at  $5  each ;  how  many 
sheep  did  I  get  ?  Ans.  252  sheep. 

28.  Sold  275  acres  of  land  at  $24  per  acre,  340  acres  at 
$32  per  acre,  and  invested  the  proceeds  in  government 
land  at  $2  per  acre ;  how  many  acres  did  I  get. 

Ans.  8,740  acres. 

29.  Of  what  number  is  36  both  divisor  and  quotient  ? 

Ans.  1,296. 

30.  Bought  12  horses  at  $80  per  head  ;  at  what  price 
must  I  sell  each  horse  so  as  to  gain  $72.         A^is,  $86. 

31.  A  party  of  45  men  has  provisions  sufficient  for  40 
days;  how  long  would  it  last  1  man  ?  2  men?  4  men?  5 
men  ?  20  men  ? 

Ans.  1  man,  1,800 days;  2  men,  900  days;  4 men,  450 
days;  5  men,  360  days;  20  men,  90  days. 

32.  A  garrison  of  900  men  have  provisions  for  6  months; 
how  many  men  would  the  same  provisions  last  9  months? 

Ans.  600  men. 

33.  If  14  men  do  a  piece  of  work  in  3  days,  how  many 
men  would  do  it  in  1  day  ?  how  many  in  2  days  ?  in  6 
days  ? 

Ans.  42  men  in  1  day,  21  men  in  2  days,  7  men  in  6 
days. 

34.  If  4  men  mow  a  field  of  grass  in  3  days,  how  long 
will  it  take  1  man  ?  2  men  ?  3  men  ?  6  men  ? 

Ans.  1  man,  12  days;  2  men,  6  days;  3  men,  4  days; 
6  men,  2 


Factoring, 


Any  number  that  is  the  product  of  two  or  more  num- 
bers is  a  Composite  Ntmiber  ;  as,  12  is  the  product 
of  4  and  3,  and  4  is  the  product  of  2  and  2  ;  hence,  the 
factors  of  12  are  2,  2,  and  3 ;  these  factors  cannot  be 
reduced,  they  are  therefore  called  Prime  Factors^  as 
any  number  is  called  Prime  which  is  not  formed  by 
other  factors  than  itself  and  unity.  There  are  sixteen 
Prime  Numbers  in  the  first  fifty ;  viz.,  1,  2,  3,  5,  7,  11, 
13,  17,  19,  23,  29,  31,  37,  41,  43,  and  47,  and  they  are 
found  thus : 

5,      0,      7,      %,      0,    10, 

n,  10,  17,  n,  19,  n, 

t$,  n,  n,  n,  29,  $0, 


1, 

% 

3, 

i, 

11, 

n, 

13, 

U, 

n, 

n. 

23, 

u, 

31, 

n, 

%t, 

u. 

41, 

it, 

43, 

u. 

$$,    $0,    37,     $$,    $%    0, 
0,    ^6,     47,    4$,    40,     $0. 

Every  even  number  after  2  is  composite,  as  it  is  divisi- 
ble by  2  ;  strike  these.  Every  third  number  after  3  is 
divisible  by  3 ;  strike  these.  Every  fifth  number  after  5 ; 
every  seventh  number  after  7 ;  the  ninth  numbers  are 
canceled  by  3 ;  every  eleventh  number  after  11,  etc. 

CoR.  1. — Every  even  number  is  divisible  by  2. 

CoR.  2. — Every  number  whose  last  two  figures  express 
a  number  which  is  a  multiple  of  4,  is  divisible  by  4 ;  for 
if  the  number  expressed  by  these  two  figures  is  subtracted 
from  the  whole  number,  the  remainder  will  be  a  certain 
number  of  hundreds  which  is  divisible  by  4. 

CoR.  3. — Every  number  ending  in  5  is  divisible  by  5. 

CoR.  4. — Every  number  ending  in  zero  is  divisible 
by  10,  consequently  by  2  and  5. 


FACTORING,  49 

CoE.  5, — Every  number  is  divisible  by  3,  when  the  sum 
of  its  figures  taken  as  units  is  divisible  by  3  ;  for  if  from 
1000  one  be  subtracted,  the  remainder  is  divisible  by  9 ; 
if  from  100  one  be  subtracted,  the  remainder  is  divisible 
by  9  ;  so  also  of  10  ;  hence  if  from  2000  two  be  subtract- 
ed, the  remainder  is  divisible  by  9  ;  so  also  take  2  from 
200,  and  2  from  20,  etc.  ;  therefore,  in  dividing  any  num- 
ber of  thousands,  hundreds,  or  tens,  the  remainder  will 
always  be  the  unit  of  thousands,  hundreds,  or  tens,  and 
if  the  sum  of  these  as  units  and  ialso  of  the  units  of  the 
given  number  equals  9,  or  any  number  of  nines,  the 
whole  number  is  divisible  by  9,  and  consequently  by  3. 


PROBLEMS. 

Resolve  the  following 

numbers  to  their  prime  fact( 

1. 

4 

= 

2,2. 

13. 

22  =  2, 11. 

2. 

6 

= 

2,3. 

14. 

24  =  2,  2,  2,  3. 

3. 

8 

= 

2,  2,  2. 

15. 

26  =  2, 13. 

4. 

9 

= 

3,3. 

16. 

27  =  3,  3,  3. 

5. 

10 

= 

2,5. 

17. 

28  =  2,  2,  7. 

6. 

13 

— 

2,  2,  3. 

18. 

30  =  2,  3,  5. 

7. 

14 

= 

2,7. 

19. 

32  =  2,  2,  2,  2,  3. 

8. 

15 

= 

3,5. 

20. 

33  =  3, 11. 

9. 

16 

^ 

2,  2,  2,  2. 

21. 

34  =  2, 17. 

10. 

18 

z± 

2,  3,  3. 

22. 

35  =  5,  7. 

11. 

ao 

= 

2,  2,  5. 

23. 

36  =  2,  2,  3,  3. 

12. 

21 

= 

3,7. 

24. 

38  =  2;  19. 

50 


1 

FACTORING 

• 

EXAMPLES. 

Resolve  the 

following  numbers  into  their 

prime  f 

actors 

1. 

60. 

10. 

86. 

19. 

106. 

28. 

135. 

2. 

64. 

11. 

88. 

20. 

108. 

29. 

128. 

3. 

65. 

13. 

90. 

21. 

110. 

30. 

130. 

4. 

70. 

13. 

95. 

22. 

112. 

31. 

132. 

5. 

73. 

14. 

96. 

23. 

114. 

32. 

136. 

6. 

75. 

15. 

98. 

24. 

116. 

33. 

140. 

7. 

78. 

16. 

100. 

25. 

iia 

34. 

325. 

8. 

80. 

17. 

103. 

26. 

120. 

35. 

500. 

9. 

84. 

18. 

104. 

27. 

124. 

36. 

635. 

LEAST    COMMON    MULTIPLE. 

Def. — One  number  is  called  a  Wiiltijjle  of  another 
when  it  is" exactly  divisible  by  that  other  number. 

When  a  number  is  resolved  into  its  prime  factors,  the 
original  number  is  a  multiple  of  all  the  prime  factors, 
and  of  all  the  quotients  arising  from  these  factors  divided 
into  the  original  number;  when  the  same  factor  occurs 
several  times,  the  different  products  of  this  factor  multi- 
plied by  itself  must  also  be  divided  into  the  original 
number. 

PROBLEMS. 

Find  the  numbers  of  which  the  following  are  multiples: 

1.  24.  Ans.  t,  ty  2,  3,  4,  6,  8,  12. 

2.  25.  Ans.  5,  5. 

3.  30.  Ans.  2,  3,  5,  6,  10,  15. 

4.  32.  Ans.  t,  %,.  %  %  2,  4,  8, 16. 

5.  36.  '  Ans.  t,  2,  $,  3,  4,  6,  9,  12, 18. 


FACTORING*  51 

6.  40.  Ans.  t,  t,  2,  5,  4,  8,  10,  20. 

7.  48.  Ans.  %  t,  %  %  3,  4,  6,  8, 12,  16,  24. 

8.  54.  Ans.  2,  3,  $,  $,  6,  9,  18,  27. 

9.  56.  Ans.  $,  t,  2,  7,  4,  8,  14,  28. 

Rem. — First  find  the  prime  factors,  then  the  quotients  arising 
by  dividing  each  prime  factor  and  the  different  products  of  these 
factors  into  the  original  number. 

PROBLEMS. 

1.  Find  the  least  common  multiple  of  8  and  12. 

8        and        12  It  is  evident  that  any  number 

2,  2,  2.  2,  2,  3.        which  contains  all   the  prime 

2x2x2x3  =  24.  factors  of    each   number  is  a 

common  multiple  of  the  given 

numbers ;  and  the  least  common  multiple  must  contain 

these  factors  and  no  others. 

CoR. — Any  factor  must  enter  the  L.  C.  M.  as  often  as 
it  does  any  given  number. 

2.  Find  the  least  common  multiple  of  6  and  15. 

6        and        15  As  3  is  common  to  both  num- 

2,  3.  3,  5.        bers,  it  must  be  taken  but  once. 

The  L.  0.  M.  is  2  X  3  x  5=30. 

3.  Find  the  least  common  multiple  of  6  and  12. 

6        and        12  As  12  is  a  multiple  of  6,  it  is 

2,  3.  2,  2,  3.      evident  that  it  contains  all  the 

prime  factors  of  6 ;  hence,  when- 
ever one  of  the  given  numbers  is  a  multiple  of  another 
given  number,  th^t  other  number  need  not  be  considered, 
and  may  be  canceled. 


2 

<J 

8 

10 

13 

15 

2 

4 

0 

6 

15 

2 

$ 

15 

52  FACTORING. 

EXAM  PLES. 

1.  Find  the  least  common  multiple  of  6, 15,  and  42. 

0    and    15    and    42  As  42  is  a  multiple  of  6, 

3,  5.  2,  3,  7.        the  6  may  be  canceled. 

Only  3  is  common. 
2  X  3  X  5  X  7  =  210  =  L.  C.  M. 

2.  Find  the  least  common  multiple  of  6,  8,  10,  12,  15. 

As  12  is  a  multiple  of 

6,  cancel  the  6  ;  and  as  2 
is  a  common  factor  of 

X  15  =  120  ^^^  ^^  ^^"^^    numbers, 

divide  the  2  into  them, 
reserving  the  divisor,  the  quotients,  and  the  numbers  not 
divided;  as  15  is  a  multiple  of  5>  cancel  5 ;  and  as  2  is 
now  a  common  factor  of  two  of  the  quotients,  divide  it 
into  them,  reserving  as  before  ;  and  as  15  is  now  a  mul- 
tiple of  3,  cancel  3 ;  the  product  of  all  the  divisors  and 
of  the  remaining  quotients  and  original  numbers,  if  there 
be  any,  will  be  the  least  common  multiple. 

3.  Find  the  least  common  multiple  of  4,  6,  8,  and  24. 
Since  24  is  a  multiple  of  each  of  the  other  numbers, 

it  is  a  common  multiple  of  all  the  numbers. 

4.  Find  the  least  common  multiple  of  6,  8,  and  10. 

Ans.  120. 

5.  Find  the  least  common  multiple  of  10,  12,  and  14. 

Ans.  420. 

6.  Find  the  least  common  multiple  of  9,  12,  18,  27, 
and  36.  Ans.  108. 

7.  Find  the  least  common  multiple  of  7,  14,  18,  21, 
and  28.  Ans.  252. 

8.  Find  the  least  common  multiple  of  5,  15,  20,  30, 
and  60,  Ans.  60. 


FACTORING,  53 


GREATEST    COMMON    DIVISOE. 

The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  largest  number  that  will  exactly  divide 
all  the  numbers. 

PROBLEM. 

Find  the  greatest  common  divisor  of  4  and  10.  Ans.  2, 

2,  2.      2,  5. 

The  greatest  common  divisor  of  any  number  of  terms, 
must  be  the  factor,  or  the  product  of  the  factors,  common 
to  the  terms. 


EXAMPLES. 

1.  Find  the  G.  C.  D.  of  6  and  15.                  Ans.  3. 

2,  3.     3,  5. 

2.  Find  the  G.  C.  D.  of  12  and  18.      Ans.  3  x  3=:a 

2,  2,  3.  2,  3,  3. 

3.  Find  the  G.  C.  D.  of  43  and  70. 

Short  method,        3 

43      70 

7 

31      35 

3        5        ^Ms,  3x7=14. 

CANCELLATION. 

THEOREM. 

The  dividend  contains  all  and  exactly  the  same 
factors  as  the  divisor  and  quotient. 

Any  composite  number  is  the  product  of  all  its  prime 
factors,  and  may  be  resolved  into  them.  The  product 
of  any  two  integral  numbers  is  a  composite  number  and 
must  contain  all  the  factors  of  both  numbers ;  and  as-  a 


54  FACTORING. 

dividend  is  the  product  of  its  divisor  and  quotient,  it 
must  contain  the  same  factors  as  its  divisor  and 
quotient. 

Cor.  1. — The  same  is  true  if  one  or  both  divisor  and 
quotient  be  fractional ;  for  when  reduced  to  a  common 
denominator,  their  numerators  •  may  be  regarded  as 
integral. 

CoR.  2. — Every  factor  of  the  divisor  will  cancel  the 
same  factor  in  the  dividend. 

Cor.  3. — The  factors  which  are  not  canceled  by  those 
of  the  divisor  will  be  the  factors  of  the  quotient 

CoR.  4. — Canceling  a  factor  in  the  dividend  divides 
the  quotient  by  the  same  factor. 

CoR.  5. — Canceling  a  factor  in  the  divisor  multiplies 
the  quotient  by  the  same  factor. 

PROBLEMS. 

1.  Divide  648  by  36. 

648  ^  t,  WW^^  ^-  =  18,  Ans. 

2.  Divide  625  by  125. 

g|  ^  M^!l^  =  5,  Ans. 
125  0,  ^,  0. 

3.  Divide  500  by  100. 

100 

4.  A  man  bought  30  yards  of  cloth  at  $5  a  yard  ;  he 
then  exchanged  it  for  other  cloth  at  $3  a  yard.  How 
many  yards  of  the  latter  did  he  get  ? 


FACTORING,  55 

— - —  ==  50  yards.  The  3  of  the  divisor  is  canceled 
into  30  of  the  dividend. 

5.  Sold  48  cattle  at  $60  each,  and  invested  the  proceeds 
in  sheep  at  $5  each  ;  how  many  sheep  were  purchased. 

^^i?  =  576  sheep. 

6.  A  farmer  sold  150  bushels  of  wheat  at  125  cents  per 
bushel,  and  invested  the  proceeds  in  oats  at  25  cents  per 
bushel ;  how  much  oats  did  he  get  ? 


Xt$  X  150 

U 


=  750  bu.  oats. 


7.  Sold  160  acres  of  land  at  $50  per  acre,  and  bought 
another  tract  for  the  amount  of  sales,  at  $40  per  acre ; 
how  much  land  was  bought  ? 

$50  X  1:00 

■:i-^ =  200  acres. 

8.  Divide  18  x  15  x  16  x  24  x  32  by  9,  5,  8, 12,  and  16. 

=  48. 


2x3x2x2x8 


9.  Divide  100  x  102  x  96  x  45  by  102  x  100  x  16  x  9. 

3:00  X  nt  X  00  X  ^0  _  „ 
10^  X  100  X  i0  X  0  ~ 


FUACTIOJfS. 


Def.  1. — If  a  unit  or  any  other  number  is  divided  into 
equal  parts,  one  or  more  of  these  parts  is  a  fraction  of 
the  whole,  and  all  the  parts  constitute  the  whole.  If  a 
unit  is  divided  into  two  equal  parts,  each  part  is  called 
one-half,  and  is  written  \y  and  the  two  halves  constitute 
the  whole  ;  thus,  f  =  1.  If  a  unit  is  divided  into  three 
equal  parts,  each  part  is  one-third  {\)  ;  two  of  the  parts, 
f ;  and  the  three  thirds  constitute  the  whole;  thus, 
-1  =  1.  If  5  is  divided  into  two  equal  parts,  each  part  is 
(five-halves  (f);  two  of  the  parts,  -^  =  5,  etc.;  if  6  is 
divided  into  three  equal  parts,  each  part  is  f  ==  2  ;  and 
two-thirds  of  6  is  4,  etc.  Fourths,  fifths,  sixths,  etc., 
are  similarly  constructed. 

2.  When  a  unit  is  divided  into  equal  parts,  any  num- 
ber of  the  parts  less  than  the  whole,  expressed  fraction- 
ally, is  called  a  Proper  Fraction  ;  as,  |,  |,  |,  -^, 
etc.  The  quotient  is  the  same  in  division  when  the 
dividend  is  less  than  the  divisor ;  as,  -^,  |f ,  etc. ;  but 
when  the  divisor  is  less  than  the  dividend,  the  quotient 
is  called  an  Improper  Fraction;  as,  f,  -^/, 
%   etc. 

3.  When  the  division  indicated  by  an  Improper  Frac- 
tion is  performed,  and  the  divisor  is  not  contained  an 
exact  number  of  times  in  the  dividend,  the  quotient  is 
partly  integral  and  partly  fractional,  and  is  termed  a 


FRACTIONS.  57 

Mixed  Number ;   thus,  f  =  li ;    -^^  —  8| ;    and 

■V-  =  lof . 

CoE. — The  denominator  expresses  the  number  of  parts 
into  which  a  unit  or  any  other  number  is  divided,  and 
the  numerator  expresses  the  number  of  parts  of  a  unit 
taken,  or  the  number  divided. 

ExEMPLiFiCATioiq^. — If  each  half  is  divided  into  two 
equal  parts,  the  whole  number  of  parts  is  four,  and  the 
one-half  has  made  two  of  those  parts ;  hence,  ^  =  ^  ;  if 
each  half  is  divided  into  three  equal  parts,  the  whole 
number  of  parts  is  six,  and  ^  =  -1;  ,\^=z^=zf  =  ^=: 
■^jy,  etc.,  and  ^  =  ^z=^  =  -^=z^,  etc. ;  hence,  if  both 
terms  of  a  fraction  are  multiplied  by  the  same  number, 
the  value  of  the  fraction  is  not  changed,  and  by  this 
principle  fractions  are  reduced  to  a  Common  Denom- 
inator. 

\    PROBLEMS./ 

1.  Eeduce  |-  and  ^  to  a  common  denominator. 

1-3=3^     and    iilzl 

2.  Eeduce  ^,  ^,  and  J  to  a  common  denominator. 

ixQ=T29       3x4=T¥^       ^^^      Tx3=T^' 

3.  Eeduce  f  and  f  to  a  common  denominator. 

|-?=|f,     and    4-s^|6. 

Cor.  1. — The  least  common  multiple  of  all  the  denom- 
inators is  the  least  common  denominator. 

CoR.  2. — Fractions  are  reduced  to  a  common  denom- 
inator thus :  Multiply  both  terms  of  each  fraction  by 
the  quotient  obtained  by  dividing  its  denominator  into 
the  least  common  denominator ;  or,  when  all  the  denom- 


58  FRACTIONS, 

inators  are  prime  to  each  other,  multiply  both  terms  of 
each  fraction  by  all  the  other  denominators. 

Cor.  3. — An  integral  number  is  reduced  to  a  fraction 
by  multiplying  it  by  the  denominator  of  the  fraction. 

EXAMPLES. 

1.  Eeduce  |,  f ,  and  ^  to  a  common  denominator. 

^ns.  ^,  II,  and  ff. 

2.  Eeduce  \,  f,  and  f  to  a  common  denominator. 

^ns,  -^j  If,  and  ^. 

3.  Reduce  4?  h  ^^^  |  to  a  common  denominator. 

^ns.  m^  m,  and  ||f. 

4.  Eeduce  |,  -|>  I?  and  ^^  to  a  common  denominator. 

^ns.  m^,  \m,  ^Wj>  and  ^WV- 

5.  Eeduce  \y  ^,  ^-J^,  and  -^q  to  a  common  denom- 
inator. Ans.  T^o,  ^oP^,  -^fo,  and  y^. 

Exemplification. — Since  ^  =  1  —  ^=:^  =  ^,  etc., 
and  1  =  1  =  1  =  ^  =  36^,  etc.,  and  fl:|=^,  |l:i=i 
ftt=i,  etc.,  and  fl|=i,  111=^,  -Ati=i,  etc. ;  hence,  if 
both  terms  of  a  fraction  are  divided  by  the  same  number, 
the  value  of  the  fraction  is  not  changed. 

CoR. — When  both  terms  of  a  fraction  have  a  common 
factor,  it  may  be  canceled,  and  the  fraction  is  thereby 
reduced  to  its  lowest  terms. 

Rem. — The  common  factor  may  be  either  a  prime  or  a  composite 
number,  and  it  is  the  greatest  common  divisor  of  the  terms. 

EXAM  PLES. 

1.  Eeduce  -^  to  its  lowest  terms.  Am.  Atl'^. 

2.  Eeduce  W  to  its  lowest  terms.  Ans,  -J. 

3.  Eeduce  y||-j  to  its  lowest  terms.-  Ans,  ^h:. 


FRACTIONS.  59 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 

EXAMPLES. 

1.  Add  and  subtract  |  and  ^.  4  is  L.  C.  D. 

2.  Add  and  subtract  |  and  ^,        6  is  L.  C.  D. 

Arts.  Sum,  -J;  difference,  -J-. 

3.  Add  and  subtract  |  and  |.        20  is  L.  C.  D. 

Ans.  Sum,  fj;  difference,^. 

4.  Add  and  subtract  f  and  f .        30  is  L.  C.  D, 

Ans,  Sum,  |^;  difference,  -gig^. 

5.  Add  and  subtract  f  and  4-        42  is  L.  C.  D. 

Ans.  Sum,  If;  difference,:^. 

Cor. — When  the  denominators  have  no  common  fac- 
tor, then  their  product  is  the  L.  C.  D.,  and  each  numer- 
ator is  multiplied  by  all  the  denominators  except  its  own 

6.  Add  I  and  |.        L.  C.  D.,  12. 

2x4__^  3x3__^ 

3  X  4  ""  12*  4  X  3  ""  12* 

8  +  9  =  17.  Sum  z=:\i=:l^. 

7.  Add  f ,  f ,  and  f  Sum  =  -W  =  ^^f . 

As  no  two  numbers  hav6  a  common  factor,  the  L.  C.  D. 
is  the  product  of  all  the  denominators ;  and  then  as  each 
denominator  is  multiplied  by  the  other  two  denomina- 
tors, so  each  numerator  must  be  multiplied  by  the  product 
of  all  the  denominators  except  its  own. 

8.  Addf,  I,  and  3^. 

4)5      8      12 

5      2        3  4x5x2x3  =  120. 

1^  =  24.  if  i^  =15.  J^  =  10. 

.•.    5,  8,  and  12  are  the  multipliers  of  the  fractions. 


60  FRACTIONS. 

9.  Add  and  subtract  -^  and  -^^. 
25  )  275  and  350 

11  14  25x11x14  =  3850,  L. CD. 

Sum  =  -J-m,  and  difference  =  -^-^. 

10.  Add  and  subtract  3|  and  2|^. 

3f  =  3,^ 

Difference  =.  1-^.     Sum  =  6^. 

^^     .3       11  X  4       44 

^''    ^*  =  yx4  =  12 

21  ___^  x3  _  27 

"^  ~   4   X  3  "~  12 

Sum  =  fl-  =  Sli.    Difference  =  ^  =  1^^. 

Rem. — Mixed  numbers  may  be  united  by  either  of  the  above 
jnethods  ;  the  latter  is  generally  preferred. 

11.  Cast  up  the  following  account: 

cents. 

•Bought  3  pairs  hose,  at  37|-  cts.  =  112^ 

5  pocket  handkerchiefs,  at  62^  cts.  =r  312  J- 
3  pocket  knives,  at  31;^  cts.  =r    93|- 

7  lbs.  sugar,  at  12|^  cts.  =    87| 

9  inkstands,  at    6 J  cts.  =    56} 

5  qrs.  paper,  at  15|  cts.  —    76^;^ 

738f 
Paid  on  account,  549;^ 

Balance  due,  189^  cts. 

The  numerators  of  the  4ths  show  the  value,  but  those 
of  the  halves  must  be  doubled,  as  4  is  the  common 
denominator. 


FRACTIONS.  61 

MULTIPLICATION    OF    PEACTIONS. 

THEOREM. 

The  -product  of  two  proper  fractions  is  less  than 
either  fraction. 

For,  if  a  number  is  multiplied  by  one,  the  product  is 
the  same  as  the  number  multiplied.  If  the  multiplier  is 
greater  than  one,  the  product  is  greater  than  the  num- 
ber ;  and  if  the  multiplier  is  less  than  one,  the  product 
is  less  than  the  number. 

In  the  multiplication  of  two  proper  fractions,  each 
factor  is  less  than  one ;  hence  the  product  is  less  than 
either  fraction. 

PROBLEMS. 

1.  Multiply  I  by  1.  Ans.  |  x  1  =  f . 

2.  Multiply  I  by  2.  Ans.  f  x  2  =  |. 

3.  Multiply  f  by  J.  Ans.  f  x  i  =  f 

It  is  evident  that  f  multiplied  by  1  =:  f ,  that  f  x  2  is 
^,  and  f  X  1^  or  1^  time  f  is  -J ;  and,  as  alternating  the 
factors  does  not  change  the  product,  therefore,  1  x  f =1 , 
2  X  I  =  t,  and  i  X  I  =  i-. 

4.  Multiply  I  by  f  Jw5.  |  x  |  =  |. 

CoR.  1. — In  multiplying  by  a  fraction  the  numerator 
is  a  multiplier  and  the  denominator  a  divisor. 

CoR.  2. — In  the  multiplication  of  fractions,  the  pro- 
duct of  all  the  numerators  is  the  numerator  of  the 
product ;  and  the  product  of  all  the  denominators  is  the 
denominator  of  the  product. 


62  FRACTIONS. 

Cor.  3. — The  product  of  two  improper  fractions  is 
greater  than  either  fraction. 

EX  A  M  PLE  s. 

1.  Multiply  i  I,  I,  f ,  I,  4, 1, 1,  ji^,  ^,  ^. 
ixtx|x|x|x|xix|x^xi|x»  =  ^. 

By  analysis,  i  of  f  =  i,  i  of  f  =  i,  i  of  f  ==:  i,  ^  of 

f  =  i,  i  of  f  =:.  I,  I  of  I  =  i,  i  of  f  =  i,  i  of  ^^  =  3V> 

2.  Multiply  I,  If,  and  |;  thus, 

2 

I  X  ^  X  I  =  ii,  product. 

3 

3.  Multiply  -^  and  f . 

A  X  i  =  U,  product- 

4.  Multiply  35|  by  9. 

f  X  9  =  ^  =       6| 
35  X  9  =  315_ 

321f  =  Product. 

Axiom  7. — If  any  number  be  both  multiplied  and 
divided  by  the  same  number,  the  value  of  the  original 
number  is  not  changed. 

CoR.  1. — If  the  multiplier  is  greater  than  the  divisor, 
the  product  is  greater  than  the  number  multiplied,  but 
if  the  multiplier  is  less  than  the  divisor,  then  the  pro- 
duct is  less  than  the  multiplicand. 

CoR.  2. — Multiplying  the  numerator  or  dividing  the 
denominator  by  any  number,  multiplies  the  fraction  by 
the  same  number. 


FRACTIONS*  63 

5.  Multiply  37  by  8f  37 

18i 
296 

Product  =  314J 

6.  Multiply  37i  by  llf. 

37|  X  llf  =  H^  X  ^  =r  iffi.  =  440^^  =  Product. 

Rem. — When  only  one  of  the  factors  is  a  mixed  number,  they 
may  be  solved  as  the  4tli  and  5th  examples  ;  but  when  both  fac- 
tors are  mixed  numbers,  it  is  better  to  reduce  them  to  improper 
fractions  as  in  6th  example. 

7.  Multiply  14|  by  4.  13.  Multiply  3147  by  35f. 

8.  Multiply  17i  by  6.  14.  Multiply  4156|  by  2124|. 

9.  Multiply  18i  by  9.  Ih.  Multiply  l'^l^\  by  13f 

10.  Multiply  24|  by  8.  16.  Multiply  15f  by  9f . 

11.  Multiply  1456  by  15f    17.  Multiply  29^  by  13^. 

12.  Multiply  2375  by  27-J.    18.  Multiply  104^  by  20f. 

DIVISION    OF    FEACTIONS. 


PROBLEMS. 

1.  Divide  10  by  5.                 Ans.  10 

-^  5  =  Y  =  2. 

2.  Divide  6  by  3. 

Ans,  1  =  2. 

3.  Divide  3  by  3. 

Ans.  1  =  1. 

4.  Divide  1  by  2. 

Ans.  \. 

5.  Divide  1  by  3. 

Ans.  \. 

6.  Divide  2  by  3. 

Ans.  f. 

7-  Divide  3  by  4. 

A71S.   f. 

8.  Divide  5  by  4. 

Ans.  f  =  IJ. 

9.  Divide  |  by  2,  or  divide  ^  into  two  equal  parts. 

Ans.  1^3  =  f 


64  FRACTIONS. 

10.  Divide  f  by  2,  or  divide  |  into  two  equal  parts. 

Ans.  f-^2  _  1^ 

11.  Divide  \  by  2,  or  divide  ^  into  two  equal  parts. 

Ans.  ix3  =  f 

12.  Divide  f  by  \,  or  how  often  is, -J-  contained  in  f. 

^^s.  Evidently  twice. 

13.  Divide  ^  by  J,  or  how  often  is  J  contained  in  4=1. 

Ans.  Evidently  twice. 

14.  Divide  ^  by  -J-,  or  how  often  is  -J-  contained  in  ^. 

Ans.  ^  =  3  and  J  =  f ;  f  -V- 1  =  I  =  H. 

15.  Divide  \  by  J,  or  how  often  is  ^  contained  in  \. 

Ans.  i  =  -^  and  i  =  ^^;  ^  -r-  -^V  =  f 

CoR.  1. — The  numerator  of  a  fraction  is  a  dividend, 
the  denominator  a  divisor,  and  the  fraction  itself  the 
quotient. 

CoE.  2. — To  divide  a  fraction  by  a  fraction,  reduce 
both  to  a  common  denominator  and  then  divide  the 
numerator  of  the  dividend  by  the  numerator  of  the 
divisor. 

CoR.  3.— Dividing  the  numerator  or  multiplying  the 
denominator  by  any  number,  divides  the  fraction  by  the 
same  number. 

THEOREM. 

To  divide  any  numher  hy  a  fraction,  invert  the 
fractional  divisor  and  malce  it  a  multiplier. 

1st.  When  the  dividend  is  integral. 

If  the  dividend  is  multiplied  by  the  denominator  of  the 
divisor,  the  product  will  be  the  numerator  of  the  quotient, 
and  the  numerator  of  the  divisor  will  be  the  denominator. 


FRACTIONS.  65 

2d.  When  the  dividend  is  also  a  fraction. 

If  the  numerator  of  the  dividend  is  multiplied  by  the 
denominator  of  the  divisor,  the  product  will  be  the 
numerator  of  the  quotient,  and  if  the  numerator  of  the 
divisor  is  multiplied  by  the  denominator  of  the  dividend, 
the  product  will  be  the  denominator  of  the  quotient. 

PROBLEMS. 

1.  Divide  15  by  |. 

15x4=1-6/,    and    ^-r-f  =-^  z=20. 
15  X I  =  20. 

2.  Divide  ^-  by  f .  ' 

-V-xt=fi    and    fx|=:^^,    and    ^z=^-  =  ^. 

Any  number  is  divided  by  a  fraction  by  inverting  the 
fractional  divisor  and  making  it  a  multiplier. 

EXAMPLES 

1.  Divide  36  by  f .  9.  Divide  83|  by  ^. 

2.  Divide  12  by  f.  10.  Divide  15  by  ^V 

3.  Divide  16  by  ^.  11.  Divide  ^-  by  ^V 
4  Divide  f  by  f .  12.  Divide  ||  by  f . 

5.  Divide  |  by  f .  13.  Divide  |f  by  -^. 

6.  Divide  |  by  f  14.  Divide  208|  by  27|. 

7.  Divide  112i  by  21f.  15.  Divide  109^  by  29f 

8.  Divide  14|  by  9f 

Eem. — Mixed  numbers  should  be  reduced  to  improper  fractions. 
16.  Divide  5436f  by  3.  Ans.  1812f 


66  FRACTIONS. 

17.  Divide  6,478|  by  9.  Ans.  719||. 

9  )  64781 

719,  remainder  ^=:^  and  3/-^9=ff. 

18.  Divide  250f  by  12|.  Ans.  19|4. 

19.  Divide  450  by  |.  ^^5.  600. 

20.  Divide!  by  20.  Ans.  ^. 

21.  Divide  i  X  f  X  f  X 1  by  I,  ^0.  -H.  H-  -^^5.  f. 

22.  Divide  f  by  J.  ^/i5.  3. 

23.  Divide  |  by  \.  Ans.  5. 

24.  Divide  1  by  i,  i,  i,  \.  Ans.  2,  3,  4,  5. 

25.  Divide  1  by  f,  |,  |.  ^7^5.  |,  !,  |. 

26.  Divide  2  by  f ,  f,  f.  ^t^s.  |,  |,  J^-. 

CoK.  1.  Any  fraction  having  unity  for  its  numerator  is 
contained  in  1  as  many  times'as  there  are  units  in  its  de- 
nominator ;  thus,  ^  is  contained  in  1  twice,  ^  three  times, 
&c.  If  the  fraction  has  2  for  its  numerator,  it  is  con- 
tained one-half  as  many  times ;  if  the  numerator  is  3,  one- 
third  as  many  times,  &c. 

CoK.  2.  If  the  dividend  is  increased,  the  quotient  is  in- 
creased just  as  many  times ;  if  the  divisor  is  increased,  the 
quotient  is  diminished  just  as  many  times ;  the  converse 
is  also  true  in  both  cases. 

CoE.  3.  To  divide  any  number,  either  integral  or  frac- 
tional, by  a  fraction,  invert  the  divisor  and  multiply  the 
dividend  by  it. 

COMPLEX  FEACTIONS. 
When  one  or  both  terms  of  a  fraction  are  either  frac- 
tions or  mixed   numbers,   it  is   called    a   Complex 
Fraction;  thus, 

34     i     3     37i     100^    ,  IPX- 

-/'    f '     9*    zrsT'    -TKK^  etc.,  are  complex  fractions. 
Off     62J      100  ^ 


FB ACTIONS,  67 

Eem. — ^When  we  consider  that  the  numerator  of  a  fraction 
represents  a  dividend,  and  the  denominator  a  divisor,  a  complex 
fraction  is  readily  reduced  to  a  simple  fraction. 

3- 

1.  Eeduce  -^  to  a  simple  fraction. 

Rem. — As  the  denominator  is  a  divisor,  it  must  be  inverted. 

2.  Eeduce  J  to  a  simple  fraction. 

I 

I  =  i  X  f  :=  f . 

3 

3.  Eeduce  ^  to  a  common  fraction. 

I 

|=^^f  =  ¥  =  ^i. 

3.  Eeduce  -^  to  a  common  fraction. 
62|- 

5.  Multiply  -^  X  ^^  =  i|i  X  ^  =  ^px^  X 


6.  Divide|by|  =  |x|  =  4x|x|x|  =  i. 

Jxix|xi  =  f. 

In  changing  the  division  to  multiplication,  the  whole 
divisor  must  be  inverted  ;  that  is,  f  becomes  the  numer- 
ator and  f  becomes  the  denominator,  with  the  sign  of 
multiplication  ;  then  again  the  two  denominators  must  be 
inverted  ;  then  they  are  all  in  the  form  of  multiplication. 


Review. 


EEDUCTION. 

Problem. — Eeduce  f,  |,  and  4  to  a  common  denomi- 
nator. 

Since  all  the  denominators  are  prime  to  each  other,  the 
common  denominator  is  their  product;  thus,  3x5x7= 
105,  the  denominator; 

And  2x5x7=J?0;  3x3x7=63;  5x3x5z=75,  the 
numerators; 

Or  |x|x^=tVV;  |x|x^=t^^;  4x|x|=t^. 

Cor.  Each  numerator  is  multiplied  by  all  the  denomi- 
nators but  its  own. 

EXAMPLES. 

1.  Eeduce  |,  f ,  and  f  to  a  common  denominator. 

Ans.  6  is  the  L.  C.  D. ;  |,  -f-,  f. 

2.  Eeduce  ^,  f,  and  f  to  a  common  denominator. 

^/^5.  L.  C.D.,60;  |i,f|.if. 

3.  Eeduce  |,  f,  and  4  to  a  common  denominator. 

Ans.  \;f%y  trf^  TTo"' 

4.  Eeduce  f ,  3^,  and  -^  to  a  common  denominator. 

Ans.    TJWTy    TTsJy    1 2  8  7 • 

5.  Eeduce  y\,  \\,  ^,  and  ^,  to  a  common  denomina- 
tor. ^7?.^.  tVV  t¥o.  t¥o.  tVo. 

6.  Eeduce  f,  f ,  ^,  ^,  and  ^  to  a  common  denominator. 

^«*'-  n,  u,  u,  a,  u- 

7.  Eeduce  y^y,  ^fy  to  a  common  denominator. 


FRACTIONS.  69 

ADDITION. 

PROBLEMS 

1.  What  is  the  sum  of  ^  and  ^  ?  Ans.  f +  f =|-. 

2.  What  is  the  sum  of  \  and  J  ?  Ans.  ^. 

3.  What  is  the  sum  of  ^  and  ^  ?  ^;^5.  -J^. 

4.  What  is  the  sum  of  \  and  ^  ?  Ans.  \f. 
CoR.  The  sum  of  any  two  fractions  whose  numerators 

are  unity,  will  have  for  its  numerator  the  sum  of  its  de- 
nominators, and  for  its  denominator  their  product. 

5.  What  is  the  sum  of  i,  J,  and  ^  ?  Ans.  |f . 

6.  What  is  the  sum  of  ^  \,  and  ^  ?  Ans.  J|4. 
CoR.  The  sum  of  any  three  fractions  whose  numerators 

are  unity  will  have  for  its  numerator  the  sum  of  the  pro- 
ducts of  the  denominators,  taken  two  at  a  time,  and  for 
its  denominator  the  product  of  all  the  denominators. 

EXAMPLES. 

1.  Add  f ,  I,  3^,  and  ^.  Ans.  2f|  _  _ 

2.  Add  f,  I,  I  -^,  and  H-  ^^s.  4^^, 

3.  Add  H,  if.  Ans.liU' 

4.  Add  if,  H-  ^^^-  hh 

5.  Add  f ,  -j^,  ^V,  and  l-J-.  -4w5.  ly^. 

6.  Add  I,  f ,  3^,  A.  Ans.  2^, 

7.  Add  I,  ^,  i^,  ^V  -i^^.  IH- 

8.  Add  i,  ^,  i,  i,  i,  T^^,  ^=  ^/i5.  If =li 

9.  Add  2^,  3|,  4^,  and  5|.  ^^5.  16^, 

10.  Add  12f ,  15f,  and  25f  ^7^s.  54|f. 

11.  Add  i,  I,  f,  and  f.  Ans.  2^, 

12.  Add  3^,  ^,  ii,  and  if.  ^7i5.  1^ 

13.  Add  \,  I,  and  ^.  ^/^5.   tf|, 
14  Add  ^  and  ^V-  ^^5-  i^^ 


70  FRACTIONS. 


SUBTEACTIOK 

PROBLEMS. 

1.  From  f  take  ^.    Ans.  ^.     4.  From  f  take  f .   Ans,  ^. 

2.  From  |  take  f.  Ans,  ^.     5.  From  ^  take  f .  Ans.  -^^» 

3.  From  f  take  |.  Ajis.  gV     ^'  Fi'om  -^  take  f.  ^^5.  -^^. 
Cor.  Two  fractions  whose  numerators  differ  by  unity, 

and  each  denominator  is  greater  by  unity  than  its  numer- 
ator, the  fraction  having  the  larger  terms  is  the  greater, 
and  their  difference  has  unity  for  its  numerator,  and  the 
product  of  the  two  denominators  for  its  denominator. 


EXAMPLES. 

1. 

From  f  take  }. 

Ans.  J. 

2. 

From  1^  take  }. 

Ans.  |. 

3. 

From  5|  take  f 

Ans.  m. 

4. 

From  5i  take  2|. 

Ans.  2J. 

5. 

From  51  take  3f. 

Ans.  2^V 

6. 

From  5f  take  4|-. 

Ans.  1^. 

7. 

From  ^  take  4f . 

J[^5.  2^. 

8. 

From  l^  take  5|. 

^7^5.  7^. 

9. 

From  15f  take  12|. 

Ans.  3t^. 

10. 

From  37f  take  21f . 

^7^5.  16t^. 

11. 

From  45|  take  34^. 

Ans.  llyV- 

12. 

From  271  take  15|. 

^^5.  llf^. 

13.  From  a  piece  of  cloth  containing  16f  yards,  a  mer- 
chant sold  7f  yds.;  how  much  was  left  ?    Ans.  8|^  yd. 

14.  A  merchant  has  2  pieces  of  muslin,  the  one  has  41f 
yards,  and  the  other  43|  yd.;  he  sells  61|-yd.;  how  much 
has  he  left  ?  Ans.  231^. 

15.  A  man  earned  $36|,  and  spent  $15| ;  how  much 
had  he  left?  Ans.  $21^. 


FB  ACTIONS,  71 

16.  James  had  $45|,  and  lost  $16|;  how  much  had  he 
left?  Ans.  $28i-f. 

17.  Subtract  9^^  from  lO^^.  Ans.  f|}. 

18.  Subtract  15-|  from  25|.  Ans.  10-^. 

19.  Subtract  36f?-  from  451^.  Ans,  9^^^. 

20.  Subtract  56|  from  97|.  Ans.  40^^. 

21.  Subtract  21f  from  35-|.  ^^5.  l^^. 

22.  Subtract  105t»j- from  205t^.  ^/^5.  99^^. 

ORAL    EXERCISES. 

Whatisjof  f  ?  ioff?  Jof  f?  iof  f  ?  ioff?  |  of 
f?ioff?   JofA? 

One-third  of  61s  i  of  what?  i?  i?  i?  ^P  -J?  i?  -^i^? 

One-third  of  9  is  i  of  what?  i?  i?  i?  4^?  i?  i?  ^? 

A?  A? 

Two-thirdsof  9  is^of  what?  i?  i?  i?  ^?  i?  i? 

^?  1^?  1^^? 

One-third  of  12  is  i  of  what?  i?  i?  i?  |?  i?  i? 

iV?  1^?   1^? 

Two-thirdsof  12isiofwhat?  J?  |?  |?  ^?  ^?  j? 

What  part  of  5  is  1  ?  2  ?  3  ?  4?  Ans.  |,  |,  |,  |. 

What  part  of  b\  is  ^  ?  H  ?  2|  ?  3^  ? 

^^5.  _^,  ^3^,  ^,  ^. 

What  part  of  o|  is  i?  IJ  ?  2^?  3^? 

Ans.  -g^,  ^,  -f^,  f^. 
What  part  of  5i  is  i ?  f  ?  4?  2^? 

What  part  of  100  is  6i  ?  13^  ?  25  ?  37^  ? 

Am.  -^y  \,  \,  |. 


72  FRACTIONS. 

What  part  of  24  hours  is  5  h.?  6  h.?  8^  h.?  9^  h.? 

Ans.  ^,  I,  H,  1^. 
What  part  of  1  is  ^?  ^  ?  J?  ^  ?  Ans.  i  i,  h  \. 
What  part  of  |  is  ^  ?  i  ?  i  ?  i?      Ans.  |,  i,  i,  ^. 

PROBLEMS. 

1.  Multiply  1  by  2. 

2.  xMultiply  1  by  1. 

3.  Multiply  1  by  f 

4.  Multiply  1  by  J. 

5.  Multiply  1  by  |. 

6.  Multiply  ^  by  f. 

7.  Divide  2  by  2. 

8.  Divide  2  by  1. 

9.  Divide  2  by  i. 

10.  Divide  2  by  J. 

11.  Divide  2  by  |. 
'     12.  Divide  ^  by  |. 

13.  If  i  cord  of  wood. cost  $2,  what  will  1  cord  cost? 

Ans.  2xf=$4. 

14.  If  i  cord  of  wood  cost  $1,  what  will  1  cord  cost  ? 

Ans.  lxf=$4. 

15.  If  f  cord  of  wood  cost  $3,  what  will  1  cord  cost  ? 

Ans.  3x|=S4. 

16.  Iff  ton  of  coal  cost  $5,  what  will  1  ton  cost  ? 

Ans.  5x|=$6. 

17.  If  ^  ton  of  coal  cost  17,  what  wiU  1  ton  cost  ? 

Ans.  7xf=$8. 

18.  If  ^  cord  of  wood  cost  $2,  what  will  3  cords  cost  ? 

Ans.  2x-|x3=$12. 

19.  If  i  cord  of  wood  cost  $1,  what  will  5  cords  cost  ? 

A71S.  4xf  xf=$20. 


Ans. 

1x2= 

=3. 

Ans. 

1x1= 

=  1. 

Ans. 

lxi= 

=i- 

Ans. 

lxi= 

=h 

Ans. 

lx|= 

=f- 

Ans. 

ixf= 

=|. 

Ans. 

2-^2= 

=1. 

Ans. 

3^1= 

=3. 

Ans. 

3xf= 

=4. 

Ans. 

2xt= 

=8. 

Ans.  3  X 1 

=^= 

2|. 

Ans  |-x 

i=i= 

=1- 

f 


FRACTIONS.  73 

20,  If  f  cord  of  wood  cost  $3,  what  will  6  cords  cost  ? 

Ans.  3x^x6=124. 

21.  If  3  cords  wood  cost  $15,  what  will  8  cords  cost  ? 

Ans.  15-r-3x8=$40. 

CoK.  1.  When  a  given  number  of  articles,  each  of  the 
same  value,  cost  a  certain  sum  of  money,  in  order  to  find 
the  cost  of  one  article,  divide  the  certain  sum  by  the 
given  number,  whether  integral  or  fractional. 

CoE.  2.  Multiply  the  cost  of  one  article  by  any  re- 
quired number  of  articles,  and  the  product  will  be  the 
cost  of  that  number. 

EXAMPLES, 

1.  If  4^  of  a  yard  of  cloth  cost  40  cents,  what  will  a  yard 
cost  ?  what  will  |  of  a  yard  cost  ?  f  ?  t^j  ?  ii  ? 

Ans.  280  cents,  186|-  cts.,  210  cts.,  100  cts.,  110  cts. 

2.  At  $9  a  barrel,  what  will  f  of  a  barrel  of  flour  cost  ? 

Ans.  $6. 

3.  At  $8  a  ton,  what  will  f  of  a  ton  of  coal  cost  ? 

Ans.  $6. 

4.  If  f  of  a  yard  of  linen  cost  18  cts.,  what  will  1  yard 
cost?  Ans.  45  cents. 

5.  If  f  of  a  yard  of  linen  cost  21  cts.,  what  will  1  yard 
cost  ?  Ans.  49  cents. 

6.  If  ^  of  a  yard  of  linen  cost  36  cts.,  what  will  \  yard 
cost  ?  Ans.  10  cents. 

7.  If  f  of  a  yard  of  linen  cost  15  cts.,  what  will  1  yard 
cost?  •  Ans.  20  cents. 

8.  If  f  of  a  yard  of  linen  cost  24  cts.,  what  will  f  yard 
cost  ?  Ans.  dO  cents. 

9.  If  f  of  a  yard  of  cloth  cost  $f ,  what  will  1  yard  cost  ? 
2  yd.?  3  yd.?  5  yd.?  Ans.  %1\,  $2^,  $3|-,  $5f. 

4 


FRACTIONS. 

10.  If  I  of  a  ton  of  hay  cost  $7,  what  will  1  ton  cost? 
gftons?   3|?   7i?   9J?     ^/^5.  $8,  $22,  $29,  $60,  $74. 
il.  What  is  the  value  of  ix|x|xt-r-fx4x|xf. 

Ans.  ^. 

12.  What  is  the  number  whose  factors  are  \^  and  -^  ? 

Ans.  f. 

13.  The  divisor  is  f  and  the  quotient  f ;   what  is  the 
dividend?  Ans,  f. 

14.  In  what  time  will  a  man  make  the  round  of  the 
earth — 24,894  miles— by  traveling  33^  miles  a  day  ? 

Ans.  746f^days. 

15.  In  one  mile  there  are  1,760  yards ;  how  many  yards 
in  5|  miles?  7^  miles?  9|  miles  ?  12f  miles  ? 

A71S.  10,120  yds.,  12,540  yds.,  16,896  yds.,  22,586|  yds. 

16.  In  one  square  mile  there  are  640  acres;  how  many 
acres  in  3|  sq.  m.?  5|  sq.  m.?  9|  sq.  m.?  12^  sq.  m.? 

A71S.  2,304  A.,  3,680  A.,  6,000  A.,  7,786f  A. 
4^17.  If  f  of  a  ton  of  coal  cost  $9,  what  will  15J  tons 
cost?  Ans.  $184. 

/C 18.  If  f  of  a  ton  of  coal  cost  $4,  what  will  31|  tons  cost? 

Ans.  $441. 

19.  Divide  f^  by  |,  and  multiply  the  quotient  by  24|. 

Ans.  14^f. 

20.  Divide  ^  by  f|,  and  then  multiply  by  ^. 

Ans.  \. 

21.  Eeduce  |  to  a  simple  fraction.  Ans.  f|. 

22.  Eeduce  |— -|  to  a  simple  fraction.         Ans.  ffj-. 

23.  Eeduce  | — |  to  a  simple  fraction.         Ans.  2^. 

T        6 


FRACTIONS. 


4      4 

24.  Eeduce  |  x  f  to  a  simple  fraction.  Ans.  |. 

25.  Eeduce  Tq -7---  to  a  simple  fraction.        Ans.  f-i. 


PRACTICAL     EXAMPLES. 

1.  A  man  owns  -j^  of  a  gold  mine,  and  sells  f  of  his 
share  for  $50,000.  What  is  the  whole  mine  worth  at  that 
rate?  Ans,  $400,000. 

2.  The  distance  from  Baltimore  to  Philadelphia  is  97 
miles;  A  starts  from  Baltimore  to  Philadelphia  at  the 
same  time  that  B  starts  from  Philadelphia  to  Baltimore ; 
A  travels  6^  miles  per  hour,  and  B  7^  miles.  In  what 
time  will  they  meet,  and  how  far  will  each  have  traveled  ? 

3.  What  number  multiplied  by  |  will  give  a  product  of 
3^?  Ans.  ^. 

4.  What  number  divided  by  f  will  give  a  quotient  of  ^  ? 

Ans.  f. 

5.  If  a  man  travel  237^  miles  in  21f  hours,  how  far 
does  he  travel  each  hour  ?  Ans.  lliVg\. 

6.  If  a  traveler  perform  a  journey  in  17f  hours,  by 
traveling  6f  miles  in  an  hour,  what  is  the  length  of  the 
journey?  Ans.  11  If  miles. 

7.  What  number  multiplied  by  ^  of  f  of  f  of  ^  will 
give  a  product  of  3^  ?  Ans.  If. 

8.  What  cost  15|  yards  of  cloth  at  $2|  per  yard  ? 

Ans.  $41^. 

9.  A  man  owning  f  of  a  farm  sold  f  of  his  share ;  what 
part  did  he  sell,  and  what  part  had  he  left  ? 

Ans.  He  sold  ^  and  had  left  ■^. 


76  FRACTIONS, 

10.  James  is  12  years  old,  and  |  of  his  age  is  9  years 
less  than  that  of  his  brother;  how  old  is  his  brother? 

A71S.  18  years. 

11.  If  8^  yards  of  muslin  at  9  cents  a  yard  is  worth  f  of 
a  gallon  of  syrup,  what  is  a  gallon  of  syrup  worth  ? 

Ans.  90  cents. 

12.  If  f  of  a  yard  of  linen  at  70  cts.  a  yard  costs  as 
much  as  ^  of  a  yard  of  muslin,  what  will  a  yard  of  muslin 
cost  ?  Ans.  25  cents. 

13.  If  a  man  build  f  of  a  rod  of  wall  in  a  day,  how 
many  rods  will  he  build  in  11 J  days  ? 

Ans.  6|-  rods. 

14.  A  and  B  bought  a  barrel  of  flour;  A  paid  $3 J,  and 
B  $4| ;  what  part  should  each  get  ? 

Ans,  A  If,  and  B  ||. 

15.  In  an  orchard  ^  of  the  trees  bear  apples,  J  peaches, 
\  plums,  and  2^  are  pear  trees;  how  many  trees  in  the 
orchard?  Ans.  120  trees. 
|<16.  One-fourth  of  a  pole  is  in  the  mud,  \  in  the  water, 
and  22  feet  above  the  water;  what  is  the  length  of  the 
pole  ?  Ans,  40  feet. 

17.  What  is  the  difference  between  f  of  ^  and  i  of  ^? 

Ans.  -gV. 

18.  A  owns  1^  of  a  gold  mine,  and  B  ^  of  the  same;  A 
sells  f  of  his  share  and  B  ^  of  his  to  C  ;  what  is  now  the 
part  of  each  ?  Ans.  A  has  \y  B  -J-,  and  C  ^. 

19.  A,  B,  and  C  purchased  160  acres  of  land;  A  is  to 
have  i  of  it,  which  is  |  as  much  as  B's,  and  0  the  balance ; 
how  much  is  C's  part  ?  Ans.  60  acres. 

"^20.,  My  horse  is  worth  $150,  which  is  |  the  value  of  the 
Duggy;  what  is  the  buggy  worth  ?  Ans.  $225. 


FRACTIONS.  '^>i^  77 

21.  A  can  do  a  piece  of  work  in  4  days  and  B  in  5  days ; 
how  much  will  A  and  B  do  in  one  day  ?  How  long  will 
it  take  both  to  do  it  ? 

Ans.  They  will  do  ^  in  1  day,  and  in  2|  days  the  whole. 

22.  A  can  do  a  piece  of  work  in  12  days,  and  B  in  15 
days ;  how  long  will  it  take  both  to  do  it  ? 

^  Ans,  6f  days. 

23/Si  cistern  has  two  pipes  for  supply  and  one  for  dis- 
charge ;  by  the  first  it  would  be  filled  in  6  hours,  and  by 
the  second  in  8  hours,  and  the  third  would  empty  it  in  4 
hours ;  in  what  time  will  it  be  filled  if  all  run  together  ? 

Ans,  In  24  hours. 

24.  How  does  multiplying  the  numerator  of  a  fraction 
affect  its  value  ? 

25.  How  does  diyiding  the  numerator  of  a  fraction 
affect  its  value  ? 

26.  How  does  multiplying  the  denominator  of  a  frac- 
tion affect  its  value  ? 

27.  How  does  dividing  the  denominator  of  a  fraction 
affect  its  value  ? 

28.  What  effect  has  multiplying  both  terms  of  a  frac- 
tion by  the  same  number  ? 

29.  What  effect  has  dividing  both  terms  of  a  fraction 
by  the  same  number  ? 

30.  To  what  term  in  division  does  the  numerator  cor- 
respond ?  the  denominator  ?  the  fraction  itself  ? 

31.  A  can  do  a  piece  of  work  in  7  days;  B  in  9  days ; 
how  much  will  each  do  in  1  day  ?  how  much  will  both 
do  in  a  day,  and  how  long  will  it  take  both  to  do  it  ? 

Ans.  A  will  do  |  and  B  -|  in  a  day ;  both  will  do  ^|  in 
a  day,  and  it  will  take  both  ^\\  days.  f\A 


78  FRACTIONS, 

32.  A  man  who  had  purchased  a  house,  found  that 
after  he  had  paid  ^  and  \  of  the  cost  he  still  owed  $500  ; 
what  was  the  cost  of  the  house  ?  Ans.  $3,000. 

33.  A,  B,  and  C  purchase  a  farm ;  A  pays  ^,  B  ^, 
and  0  the  balance,  which  was  $4,700 ;  how  much  did  A 
and  B  each  pay  ?  Ans.  A,  $3,000,  B,  $3,300. 

34.  A  and  B  together  have  $170,  and  f  of  A's  money  is 
equal  to  f  of  B's ;  how  much  has  each  ? 

A'sx  f  =:B'sx  f 

A'sx-i^=:B'sXtV 
8  A's=9  B^s;  hence  A  has  $9  as  often  as  B  has  $8. 

Ans.  A,  $90;  B,  $80. 
rS5.  The  sum  of  two  numbers  is  56,  and  |  of  the  first  is 
equal  to  f  of  the  second ;  what  are  the  numbers  ? 

Ans,  First,  32;  second,  24. 

36.  The  product  of  two  numbers  is  33^,  and  one  of  the 
numbers  is  5^;  what  is  the  other  number  ?    Ans,  6^^. 

37.  The  factors  are  25f  and  37|;  what  is  the  product  ? 

Ans,  969^^. 

38.  At  $5|  a  ton,  how  many  tons  of  coal  can  be  bought 
for  $123|  ?  Ans,  21ff  tons. 

39.  The  weight  of  3  loads  of  hay  were:  1st,  2  T. 
5  cwt.  30  lbs.;  the  2d,  3  T.  2  cwt.  25  lbs.  and  the  3d,  2  T. 
8  cwt.  60  lbs.;  what  did  the  three  loads  weigh  ? 

Ans,  7  T.  16  cwt.  15  lbs.  or  7.8075,  or  7Hf  tons. 

40.  The  dividend  is  205f  and  the  quotient  25^ ;  what 
is  the  divisor  ?  Ans,  S-^. 

41.  My  horse  is  worth  1^  times  as  much  as  my  car- 
riage ;  both  are  worth  $250  ;  what  is  each  worth  ? 

Ans,  Horse,  $150;  carriage,  $100. 

42.  If  If  acres  of  land  cost  $87^,  what  will  12|  acres 
.cost  ?  Al71s.  $630. 


'Decimal  Fbactioj^s. 


Fractions  whose  denominators  are  10, 100,  1000,  etc., 
are  rendered  decimals  of  the  same  name  by  a  little  change 
in  form ;  thus,  a  decimal  point  is  placed  on  the  left  of 
the  decimals,  or  on  the  right  of  the  units,  and  the  same 
relation  exists  between  the  successive  orders,  as  in 
abstract  numbers,  but  the  orders  themselves  are 
reversed. 

T^  =  .1,  ii)W  =  -001, 

tU  =  .01,  ttW  =  .0001, 

and  are  read  alike;   thus, 

one  tenth,  one  thousandth, 

one  hundredth,        one  ten-thousandth. 

Also,  -^  =  .3,  read  three  tenths ; 

yfg-  =  .07,  seven  hundredths ; 
■^  =  .36,  thirty-six  hundredths ; 
^^  =  .456,   four  hundred  and  fifty-six 
thousandths. 

Hence,  to  enumerate  a  decimal  fraction,  read  it  as 
you  would  an  integral  number,  adding  to  this  the  name 
of  the  denominator,  when  a  common  fraction,  which 
will  be  expressed  by  1  with  as  many  zeros  attached  to  it 
as  there  are  numbers  of  decimal  figures. 


80 


DECIMAL    FRACTIONS. 


ADDITION  AND  SUBTKAOTION. 

EXAM  PLES 

1.  Add  .1,  .01,  .001,  .0001,  and  .00001 ;  thus  : 


(1.) 

(2.) 

(3.) 

.1 

Add 

.0234 

Add  5.634 

.01 

.213 

21.321 

.001 

.3146 

.654 

.0001 

M 

.012 

.00001 

.6 

5.364 

.11111 

(*•) 

1.4710 

32.985 
(5.) 

From 

4.36315 

FVom 

.326159 

Take 

1.83754 

Take 

.234573 

Rem. 

2.53461 

Rem. 

.091586 

Cor. — As  the  relation  of  the  orders  are  the  same,  and 
the  decimals  rise  in  value  in  the  same  direction,  whilst 
in  name  they  take  the  opposite  direction ;  hence,  addi- 
tion and  subtraction  of  decimals  are  performed  as  in 
Integral  Numbers. 

MULTIPLICATION. 

THEOREM. 

In  the  v^ultiplicatio^^  of  decimals,  the  product 
will  have  as  many  places  of  decimals  as  hath 
factoids. 

t\  X  -r^  =  tU        •••        .1  X  .1  =  .01, 
and 

^^xi^  =  T(5W      •••      -01  X  .1  =  .001. 


DECIMAL    FRACTIONS.  81 


1st  ool. 

2d  OOL. 

1x1  =  1 

and 

1   X  .1   =   .1 

.1x1=    .1 

and 

.1  X  .1  =  .01 

.01   X  1  =     .01 

and 

.01  X  .1  =  .001 

or, 


The  first  column  of  products  is  the  same  as  the  first 
column  of  multiplicands,  as  1  is  the  multiplier.  The 
multiplier  in  the  second  case  is  one-tenth,  consequently 
the  products  of  the  second  column  must  be  one-tenth  of 
the  first. 

Therefore  the  product  of  two  decimal  factors  will  have 
as  many  decimal  places  as  both  factors. 

1x1      =1,  units. 
.1  X  .1     =  .01,  hundredths. 
.01  X  .01  =  .0001,  ten  thousandths. 

1x1      =1,  units.  \ 

10  X  10    z=  100,  hundreds. 
100  X  100  =  10000,  ten  thousands. 

Rem. — Observe  the  correspondence  in  name,  when  the  contrary 
orders  are  multiplied. 

PROBLEMS. 

1.  Multiply  2.  Multiply 

3.156  .534 

.215  .136 

15780  3204 

3156  1602  . 

6312  534 

.678540  .072624 

Rem. — Each  product  must  have  six  decimals,  hence  ir.  the 
second  example  a  zero  must  be  prefixed. 


82  DECIMAL  FRACTIONS. 

(3.)  (4.) 

.01  .00001 

.01  .00001 


.0001  .000000001 

DIVISION. 

Corollaries  to  Theorem,   Page  54. 

Cor.  1. — As  the  product  of  the  divisor  and  quotient  is 
equal  to  the  dividend,  therefore  the  dividend  has  as  many 
decimal  figures  as  both  divisor  and  quotient. 

CoR.  2. — If  the  divisor  has  decimal  figures  and  the 
dividend  has  none,  or  less  than  the  divisor,  as  many  must 
be  added  to  the  dividend  as  to  make  the  number  equal  to 
that  of  the  divisor,  and  then  the  quotient  will  be  inte- 
gral. If  more  decimals  are  added  to  the  dividend,  the 
quotient  wiU  contain  as  many. 

PROBLEMS. 

1.  Divide  21.4263  l?y  3.12. 

3.12  )  21.42|63  (  6.86+  As  the  divisor  has  two 

18  72  places    of     decimals,    the 

2  706  quotient  will  be  integral 

2  496  for  two  places  of  decimals 

2103  in  the  dividend ;  after  that 

1872  the  quotient  will  be  deci- 

231,  remainder.  mal. 

2.  Eeduce  the  fraction  J  to  a  decimal. 

4  )  1.00  3 

.25  5)  3.0 

.6 
CoR. — Any  common  fraction  may  be  reduced  to  a  deci- 
mal by  performing  the  division  indicated  by  the  terms. 


DeJSTOMIJ^ATE    JSf'UMBERS. 


All  arithmetical  numbers  may  be  considered  Denom^ 
inate,  even  abstract  numbers,  as  every  figure  in  each 
successive  order,  beginning  at  the  right  and  going  to  the 
left,  is  ten  times  the  value  of  the  same  figure  in  the  pre- 
vious order,  and  may  be  arranged  in  a  table ;  thus, 

10  units  =  1  ten. 

10  tens  =  1  hundred. 

10  hundred  =  1  thousand. 

10  thousand  =  1  ten-thousand. 

In  the  United  States  currency,  the  orders  have  the 
same  relation ;  thus, 

10  mills  (m.)  =z  1  cent  {ct.). 
10  cents  =  1  dime. 

10  dimes         =  1  dollar  ($). 
10  dollars        =  1  eagle. 

Dimes  and  eagles  are  coins,  but  are  not  regarded  in 
<iomputation  ;  but  only  dollars  ($),  cents,  and  mills,  the 
cents  holding  two  places. 

There  is  generally  a  decimal  point  placed  between 
dollars  and  cents ;  thus,  $456,295,  which  is  numerated 
**  four  hundred  and  fifty-six  dollars,  twenty-nine  cents 
and  five  mills.  It  may  also  be  numerated  without  any 
change  in  its  value,  "  four  hundred  and  fifty-six  thou- 
sand, two  hundred  and  ninety-five  mills. 


84  DENOMINATE    NUMBERS. 


ADDITION. 

As  the  relations  of  the  orders  in  United  Stated 
money  is  the  same  as  in  abstract  numbers,  hence 
their  applicaMon  is  the  same;  and  in  addition 
and>  subtraction  like  orders  must  be  -placed  under 
each  other,  and  in  every  other  way  the  same 
methods  are  followed. 

PROBLEMS. 

$25,365  1.  What   is   the  sum  of  twenty-five 

12.184  dollars,  thirty-six  cents  and  five  mills; 

9.100  twelve  dollars,  eighteen  cents  and  four 

30.005  mills;  nine  dollars  and  ten  cents;  thirty 

15.030  dollars  and  five  mills;    fifteen  dollars 

$91,684  ^^^  three  cents. 

Arts.,  Ninety-one  dollars,  sixty-eight  cents  and  four 
mills. 

2.  Add  the  following  sums  of  money : 

Five  dollars,  thirty  cents  and  four  mills.  $5,304 

Three  dollars  and  two  mills     ....  3.002 

Two  dollars  and  three  cents    ....  2.030 

Seven  dollars  and  three  mills  .    .    .    .  7.003 

Twelve  dollars  and  one  cent    ....  12.010 

Nine  dollars 9.000 

«3&349 


DENOMINATE    NUMBERS.  86 


Add 


(3.) 

$97,548 
68.754 
97.633 

198.564 

(4.) 

Add    $386,946 

5372.875 

64759.654 

876943.687 

(5.) 

Add  387,643  milla 

548,753 

659,864 

3,317,634 

$463,498   . 

$947463.163 

4,713,893  mills. 

Rem.  1. — The  sum  of  the  Ikst  example  may  be  numerated  thus : 
Four  millions  seven  hundred  and  thirteen  thousand,  eight  hun- 
dred and  ninety-three  mills ;  or,  thus  :  Four  thousand  seven 
hundred  and  thirteen  dollars,  eighty-nine  cents  and  three  mills. 

Hem.  2. — Mills  are  numerated  the  same  as  abstract  numbers. 

SUBTEACTIOH. 

$287,304  1.  From  two  hundred  and  eighty- 

194.293        seven    dollars,  thirty  cents    and  four 
$93,011        mills,  take  one  hundred  and  ninety- 
four   dollars,    twenty-nine  cents    and 
thvc^  mills.    Eejnainder,  Ninety-three  dollars,  one  cent 
an^i  9ne  milL 

(2.)  (3.)  (4.) 

$475648.364     $9,486,397,213     $21795.375 

387654.875      6,397,423.875      10963.625 

$87993.489     $3,088,973,338     $10831.750 

(5.)  (6.)  (7.)  (8.)  (9.) 

100000  100  100  100.00  100.00 

99999  99  1  1.50  2.50 

1  1  99  98.50  97.50 

Rem.— As  in  addition  and  subtraction,  so  also  in  multiplication, 
the  process  is  the  same  as  that  of  abstract  integers  and  decimals ; 
he«»ie  there  is  no  need  of  further  exemplification. 


86  DENOMINATE    NUMBERS. 

English  money  is  reckoned  in  pounds,  shillings,  pence, 
and  farthings;  sometimes  also  in  guineas;  thus, 

TABLE. 

4  farthings  {far,)  =  1  penny  (d.). 
12  pence  =  1  shilling  (s.). 

20  shillings  =  1  pound  (£). 

21  shillings  =  1  guinea. 

PROBLEMS. 

Reduce  £1  to  shillings,  pence,  and  farthings. 
£1 
20  £1  =   20  shillings, 

20  =  shillings.         ^1  =  ^^0  pence. 
12  £1  =  960  farthings. 

240  =  pence. 
4 

960  =  farthings. 
As  there  are  twenty  shillings  in  one  pound,  there  will 
always  be  twenty  times  as  many  shillings  as  pounds; 
and  as  there  are  twelve  pence  in  every  shilling,  there  will 
be  twelve  times  as  many  pence  as  shillings;  and  four 
times  as  many  farthings  as  pence. 

CoE. — A  higher  denomination  is  reduced  to  a  lower 
one  by  multiplication. 

Reduce  960  farthings  to  pence,  shillings  and  pounds; 
thus,  4  )  960  farthings. 

12  )^40  pence. 
20  )    20  shilHngs. 
1  pound. 
As  four  farthings  make  one  penny,  there  will  be  one- 


DENOMINATE    NUMBERS,  87 

fourth  as  many  pence  as  farthings,  one-twelfth  as  many 
shillings  as  pence,  and  one-twentieth  as  many  pounds  as 
shillings. 

CoR. — A  lower  denomination  is  reduced  to  a  higher 
one  by  division. 

Eeduce  1095  farthings  to  pence,  shillings,  and  pounds. 

4  )  1095  farthings. 
12  )    273  ...  3  far. 

20) 22  .  .  .  9d. 

£1  2s.  9d.  3  far. 

The  first  remainder  is  farthings,  the  second  pence,  and 
the  third  shillings. 

Eeduce  £1    2s.     9d.    3  far.      to  farthings. 

20 

22  shillings. 
12 

273  pence. 
4 


1095  farthings.    ■ 

In  reducing  a  higher  denomination  to  a  lower  one, 
begin  by  multiplying  by  the  number  of  the  next  lower 
denomination  that  makes  one  of  the  higher,  and  if  it  be 
a  compound  number,  add  to  the  product  the  number  of 
the  lower  denomination,  and  continue  this  process  until 
you  reach  the  lowest  denomination. 

In  reducing  a  lower  to  a  higher  denomination,  divide 
by  the  number  of  the  lowest  denomination  that  makes 
one  of  the  next  higher,  and  if  there  be  a  remainder,  it 
will  be  of  the  lowest  denomination,  etc. 


88  DENOMINATE    NUMBERS, 

Cor. — In  the  computation  of  compound  numbers,  in- 
stead of  carrying  a  unit  to  a  higher  order  for  every  ten, 
as  in  abstract  numbers,  a  unit  is  carried  to  a  higher 
denomination  as  often  as  the  sum  : -caches  the  number 
that  it  takes  of  the  lower  denomination  to  make  one  of 
the  next  higher  denomination ;  thus,  as  4  farthings  make 
''  1  penny,  as  often  as  the  sum  of  the  farthings  reaches 
four,  one  must  be  carried  to  the  pence ;  and  as  12  pence 
make  1  shilling,  in  computing  pence  as  many  must  be 
carried  to  shillings  as  the  number  of  times  12  is  contained 
in  the  number  of  pence ;  1  from  shillings  to  pounds  for 
every  20. 

In  division,  the  order  ia  reversed,  as  then  we  begin 
with  the  highest  denomination  and  descend. 


1  Add 


24  5  11  1 

The  sum  of  the  first  column  is  9  farthings,  which  is 
2  times  4  and  1 ;  the  1  is  farthings,  and  must  be  placed 
under  the  farthings  ;  the  2  is  carried  to  the  next  denom- 
ination and  added  with  the  pence,  the  sum  of  which  is 
35;  that  is,  2  times  12  and  11,  that  is,  2  shilhngs  and 
11  pence ;  the  2  is  added  with  the  shillings,  making  the 
sum  45,  which  is  £2  5s. ;  the  shillings  are  placed  under 
the  shillings  and  the  2  carried  to  the  pounds,  the  sum  of 
which  is  24. 


EXAMPLES. 

£ 

».            d. 

fa/r. 

3 

8                 1 

3 

5 

9            6 

3 

6 

11            9 

1 

8 

15          11 

3 

Z^  d 

DENOl^ 

TIN  ATE 

AUJiBEBS,  f     ^  ^ 

£ 

S. 

d. 

far. 

54 

6 

5 

1 

28 

7 

6 

3 

89 


2.  From 


£25  18s.        lOd.         2  far. 

As  you  cannot  subtract  3  farthings  from  1  farthing, 
you  must  borrow  1  penny,  which  is  4  farthings ;  this  4 
and  the  1  make  5 ;  then  3  from  5, 2  remains ;  the  1  penny 
borrowed  must  be  carried  to  the  6,  which  makes  7,  which 
cannot  be  subtracted  from  5 ;  1  shilling,  that  is,  12  pence, 
must  be  borrowed  and  added  to  the  5,  which  makes  17 ; 
7  from  17,  10  remains ;  1  shilling  to  carry  to  7  makes  8, 
which  cannot  be  taken  from  6 ;  1  pound,  that  is,  20  shil- 
lings, must  be  borrowed  and  added  to  the  6,  making  26, 
from  which  subtract  8  and  18  remains;  and  £1  to  carry 
to  28,  making  29,  which  is  subtracted  from  54  and  25 
remains. 

Rem. — When  the  subtrahend  is  less  than  the  minuend,  the  dif- 
ference can  be  taken  directly. 


& 

8. 

d. 

far. 

3.  Multiply 

4 

6 

5 

3 

by 

5 

£31 

12s, 

4 

3 

4 

6 

5 

3 

_5 

^ 

5 

5 

21 

20)32 

25 

)15 

£1 

12s. 

•     3 

12)28 

3d.  3 

2s.  4d. 
Cob. — Multiply  each  denominate  number,  and  diyide 
the  product  by  the  number  of  this  denomination  that  it 


90  DENOMINATE    NUMBERS, 

takes  to  make  one  of  the  higher,  and  carry  the  number 
of  times  it  is  contained  to  the  higher  denomination,  and 
place  the  remainder  under  its  kind. 

4.  Multiply  £48  12s.  7d.  2  far.  by  6. 

5.  Divide        4 )  £5     6s.    3d.     1  far,    by  4. 

£1  6s.  6d.  3J^far. 
4  is  contained  in  5,  once  and  £1  over;  this  £1  is  20 
shillings,  which  added  to  the  6  shillings  make  26  shillings, 
into  which  4  is  contained  6  times  and  2  shillings  over; 
this  2  shillings  is  24  pence,  which  added  to  the  3  pence, 
makes  27  pence,  in  which  4  is  contained  6  times  and 
3  pence  over,  which  is  12  farthings,  and  1  more  make  13. 
in  which  4  is  contained  3J  times. 

6.  Divide  £754  15s.  9d.  3  far.  by  27. 

27  )  £754    15s.     9d.    3  far.  (  £27 
54 
214 
189 
25 
20 

515  ( 19s.  Add  the  15s. 
27_ 
245 
243 
2 
12 

33  ( Id.    Add  the  9d. 
27  • 

6 
_4 

27(1  far.    Add  the  3  far. 

27 

Quotient  =  £27  19s.  Id.  1  far. 


DENOMINATE    NUMBERS.  91 

7.  Multiply        £5    4s.    6d.    1  far.    by  35. 

35 


£183  18s. 

3d.    3  far. 

35 

35 

35 

4)35 

5 

175 

4 
140 

6 
310 

8d.  3  far. 

7 
£183 

18 
30)158(7 

8 
13)318 

140 

18s.  3d. 

18 

Rem. — Observe  these  solutions  carefully  j  for  if  they  are  under- 
Btood,  there  is  no  further  difficulty  in  denominate  numbers ;  the 
principle  is  the  same  in  all,  the  tables  alone  differ. 

EXAM  PLES. 

1.  In  2  dollars,  how  many  cents  ?    How  many  mills  ? 

$2x   100=    200  cents. 
2  X 1000  =  2000  mills. 

2.  In  5  dollars,  how  many  cents  ?    How  many  mills  ? 

3.  In  7  dollars,  how  many  cents?     How  many  mills? 

4.  In  5  dollars  15  cents^  how  many  cents?    How 
many  mills  ? 

$5  =  500  cents. 
15 

515  cents  =  5150  mills. 

5.  In  6  dollars  15  cents  and  3  mills,  how  many  mills  ? 

6.  In  500  cents;  how  many  dollars  ?    fg^  =  $5,  Ans. 

7.  In  625  cents,  how  many  dollars  and  cents  ? 

Ans.  16.25. 


92  DENOMINATE    NUMBERS. 

8.  In  5325  mills,  how  many  dollars,  cents,  and  mills  ? 

Ans.  $5,325. 

9.  In  63257  miUs,  how  many  dollars,  cents,  and 
mills? 

10.  In  75325  cents,  how  many  dollars  and  cents  ? 

11.  If  1  bushel  of  wheat  cost  $1,125,  what  will  8  bushels 
cost  ? 

12.  If  1  bushel  of  wheat  cost  $1.05,  what  will  10 
bushels  cost  ? 

13.  If  1  bushel  of  wheat  cost  $1.05,  what  will  100 
bushels  cost  ? 

14.  If  8  bushels  of  wheat  cost  $9,  what  cost  1  bushel? 

15.  If  8  bushels  of  wheat  cost  $9,  what  cost  35  bushels  ? 

16.  If  10  bushels  of  wheat  cost  $10.50,  what  cost  53 
bushels  ? 

17.  Bought  dry  goods  for  $243.37;  groceries  for 
$146,294;  hardware  for  $71.96;  notions  for  $21,512. 
What  was  the  amount  of  the  bill  ?  Sold  the  same  at  a 
profit  of  $157,192.     What  did  I  sell  the  whole  for  ? 

18.  If  5  lbs.  sugar  cost  50  cents,  what  will  6  lbs.  cost  ? 
7  lbs.?  8?  9?  10?  11?  12? 

19.  If  6  lbs.  cost  72  cts.,  what  wiU  7  lbs.  cost  ?  8  lbs.  ? 
9?  10?  11?  12? 

20.  In  15  farthings,  how  many  pence  ? 

Ans.  3|  pen 

21.  In  18  farthings,  how  many  pence?  How  mg-ny 
pence  in  21  far.  ?  23?  25?  27  ?  29  ?  31  ?  33?  34?  35? 

22.  How  many  shillings  in  25  pence  ?  in  28  ?  35  ?  38  ? 
45?  51?  56?  65? 

23.  How  many  pounds  in  35  shillings?  in  40?  50? 
60?  65?  70?  75?  80?  85?  90?  95?  100?  105?  110? 
120? 


DENOMINATE    NUMBERS.  93 

^  24.  How  many  farthings  in  £9  13s.  9d.  3  far.  ? 
\  25.  How  many  pounds,  shillings,  pence,  and  farthings 
in  37864321  farthings? 
ft  26.  Multiply  £4  8s.  9d.  3  far.  by  9. 
27.  Divide  £25  9s.  4d.  1  far.  by  13  ? 


AVOIRDUPOIS,  OR  COMMERCIAL  WEIGHT, 

is  used  in  commercial  transactions,  when  goods  are 
bought  or  sold  in  quantity,  and  for  all  metals  except 
gold  and  silver.  * 

TAJiLE. 

16  drams  {dr.)  =  1  ounce  {oz.) 
16  ounces    *     =  1  pound  {lb). 
25  pounds         =:  1  quarter  {qr). 
4  quarters        ~  1  hundredweight  (cwt). 
20  cwt.  --:=  1  ton  (T.). 

EXEMPLIFICATIOIT. 

1  T. 

20 

—  16  )  512000  dr. 
20  cwt.  ■- 

.  16  )  32000  oz. 

—  25  )  2000  lbs. 
80  qrs.  ^- 

25  ^  )^_  ^^s- 


2000  lbs.  ^^  )1^_  c^^ 

16  1  T. 


32000  oz. 
16 


512000  dr. 


94 


DENOMINATE    NUMBERS. 


16 


T.  cwt. 
Reduce   3    4 

ST. 

2 

8 

OS.   dr. 
6    10 

20 

64 

4 

16)1653354 

258 

16 )  103334  ...  10  dr. 

25 

25)6458  ...    6oz. 

6458 

16 

103334 

4)258  .  .  .    8  1b. 
20 )  64  .  .  .    2  qr. 
3    4    2    8    6 

10 
T.  cwt.  qr.  lb.   oz.    dr. 
1653354  drams. 

Kediice  1653354  drams  to  the  original  denominations. 


TEOY    WEIGHT 

is  used  for  gold,  silver,  and  jewels ;  also  in  philosophical 
experiments. 

TABLE. 

24  grains  {gr)       =  1  pennyweight  {pwt.y 
20  pennyweights  =  1  ounce. 
12  ounces  =  1  pound. 


lib. 

12 

24 )  5760  gr. 

12  oz. 

20  )  240  pwt 

20 

12  )  12  oz. 

240  pwt. 

1  lb. 

24 

5760  qr. 


DENOMINATE    NUMBERS.  ^5 

Eeduce       5  lb.    6  oz.     10  pwt.     16  gr. 


5  1b. 

6  oz.     10  pwt. 

12 

66 

Add  the  6  oz. 

20 

1330 

Add  the  10  pwt. 

24 

31936      Add  the  16  gr. 
Eeduce  31936  grs.  to  the  original  denominations. 
24  )  31936  gr. 
20  )  1330  .  .  .  16  gr. 
12)^  .  .  .  10  pwt. 

5  lb.  6  oz.  10  pwt.  16  gr. 

DIAMOND    WEIGHT. 

Used  for  diamonds  and  other  precious  stones. 

TABLE, 

16  parts    =  1  grain  =    .8  grain  Troy. 
4  grains  =  1  carat  =.  3.2  grains  Troy. 

APOTHECAEIES'    WEIGHT 

is  used  by  druggists  in  putting  up  prescriptions ;  the 
pound,  ounce,  and  grain  are  the  same  as  in  Troy  Weight. 

TABLE. 

20  grains      =  1  scruple  (3). 

3  scruples  =  1  dram  (  3  ). 

8  drams      =  1  ounce  (  §  ). 
12  ounces    =  1  pound. 


96  DENOMINATE    NUMBERS. 


1  lb. 

13 

12  ! 

8 

96  3 

3 

288  3 

20 

20  )  5760  gr. 

3  )  288  3 

8)96  3 

12)12  I 

1  lb. 

5760  gr. 

APOTHECAEIES    FLUID    WEIGHT 
is  used  for  liquids  in  medical  prescriptions. 

TABLE. 

60  minims  (m)  =  1  fluid  dram  (f  3  ). 
8  fluid  drams    =  1  fluid  ounce  (f  |  ). 
16  fluid  ounces   =  1  pint  (0.). 
8  pints  =  1  gallon  {Cong). 

For  ordinary  use,  1  teacup  =  2  wine  glasses  =  8  table, 
spoons  =  32  tea-spoons  =  4  f  | . 

COMPARISON-   OF  WEIGHTS. 

1  lb.  Avoirdupois  =  7000  gr.  Troy. 
1  lb.  Troy  =  5760  gr.  Troy. 

LIISTEAE    MEASUEE 
is  used  for  lengths  and  distances. 

TABLE. 

12    inches  {in.)  =  1  foot  {ft). 

3    feet  =  1  yard  {yd.). 

5^  yds.,  or  16^  ft.  =1  rod  {rd.). 

40    rods  =  1  furlong  {fur.). 

8    furlongs  =  1  mile  {m.). 

3   miles  =  1  league  {lea.). 


DENOMINATE    NUMBERS,  97 

MAEINEE^S    MEASUKE. 

6  feet  =  1  fathom. 

120  fathoms  =  1  cable  length. 

880  fath.,  or  7-J^  cable  lengths    =  1  mile. 

Rem. — 1  nautical  league  =  3  equatorial  miles  =  3.45771  statute 
miles.  60  equatorial  miles  =  69.1542  statute  miles  =  1  equatorial 
degree  (°).  360°  =  the  circumference  of  a  circle.  360  equatorial 
degrees  =  the  circumference  of  the  earth. 

CLOTH    MEASUKE. 

<C^2|^  inches  {in,)     =  1  nail  {na.). 
4    nails  or  9  in.  =1  quarter. 


4 
3 
5 
6 

quarters 
quarters 
quarters 
quarters 

=  1  yard. 
=  1  ell  Flemish. 
=  1  ell  English. 
=  1  ell  French,  y 

SUEVEYOE'S  MEASUEE  OF  LENGTH. 

25 

100 

10 

8 

r  inches                            =  1  link  {I). 
links                             =  1  pole  {p.), 
links,  4  poles,  66  feet  =  1  chain  {ch.). 
chains                            =  1  furlong, 
furlongs,  or  80  chains  =  1  mile. 

LAND 

MEASUEE. 

40  perches 
4  rods 

z=  1  rod. 
=  1  acre. 

640  acres 

=  1  square  mile, 

termed  a  Section. 

98 


DENOMINATE    NUMBERS. 


SQUAEE    MEASURE. 

A  Square  is  a  surface  bounded  by  four  equal  sides, 
its  angles  are  also  equal;  thus, 

This  figure,  ABCD,  repre- 
sents a  square  foot,  consider- 
ing each  of  the  small  spaces 
as  an  inch.  The  sides,  AB, 
BC,  AD,  and  DO,  each  equal 
to  12  inches  in  length.  The 
angles,  A,  B,  C,  and  D,  are 
equal ;  that  is,  if  one  is  placed 
on  the  other,  the  sides  respec- 
'^  °    tively  will  coincide. 

AB  is  12  inches  long,  and  every  inch  in  width  makes 
12  square  inches  ;  and  the  12  inches  in  width,  which  is 
either  AD  or  BC,  makes  12  x  12  =  144  sq.  in, ;  hence 
the  correspondence  of  linear  and  square  measure ;  thus, 

LINEAR  MEASURE. 

12  inches  =  1  foot. 
3  feet  =  1  yard. 
6^  yards    =  1  rod,  pole,  or  perch. 

SQUARE   MEASURE. 

12  X  12  =  144  square  inches  =  1  square  foot. 

3  X    3  1=       9  square  feet      =  1  square  yard. 

5^  X  5^  =  30:^  square  yards    =  1  square  rod ; 

also  called  perch. 

CUBIC   MEASURE. 

12  X 12  X 12  =  1728  cubic  inches  =  1  cubic  foot. 
3x3x3=      27  cubic  feet      =  1  cubic  yard. 


BJENOMINATE    NUMBERS.  99 

CUBIC    MEASXJEE. 

A  Cube  is  a  solid  figure  bounded  by  six  equal  squares ; 
the  square  on  page  72  represents  the  base  or  any  other  side, 
as  the  sides  are  all  equal ;  the  length  of  all  the  edges  are 
equal,  and  the  angles  are  all  equal.  If  the  above  figure 
have  12  inches  altitude  added  to  it,  every  inch  will  make 
144  cubic  inches  and  12  inches  in  altitude,  144  x  12  = 
1728,  which  is  the  number  of  cubic  inches  in  a 
cubic  foot. 

TABLE. 

V  1728  cubic  inches      =  1  cubic  foot. 

27  cubic  feet  =  1  cubic  yard. 

16  cubic  feet  =  1  cord  foot. 

128  cubic  feet,  or  )        ^     ,     «        -,  ,  *  i 
^       -,  p    I  f  =  1  cd.  of  wood,  bark,  etc. 

8  cord  feet  3 

40  cubic  feet  of  round  timber,  or  50  cubic  feet  of  hewn 
timber  =  1  Ton. . 

A  perch  of  stone  is  16|^  feet  long,  1|  feet  wide,  and 
1ft.  high  =  24|  solid  feet. 

LIQUID    MEASUEE. 

This  measure  is  used  for  all  liquids. 

TABLE. 

4  gills  {gi.)  =  1  pint  (pL). 
2  pints  =  1  quart  (qL). 

4  quarts        =  1  gallon  (gal). 

In  all  liquids,  except  ale,  beer,  and  milk,  the  gallon  is 
231  cubic  inches. 

In  ale,  beer,  and  milk,  it  is  282  cubic  inches. 


100  DENOMINATE    NUMBERS. 

Rem. — In  the  former  31J  gallons  is  called  a  barrel,  63  gallons  a 
hogshead,  42  gallons  a  tierce,  84  gallons  a  puncheon,  and  126  gal- 
lons a  pipe,  and  2  pipes  a  tun.  In  the  latter,  36  gallons  =  a 
barrel,  and  54  gallons  =  a  hogshead  ;  these,  however,  are  not 
measures,  but  only  vessels. 

DKY    MEASUEE 
is  used  for  grain,  fruits,  vegetables,  coal,  salt,  etc. 

TABLE. 

2  pints     =  1  quart. 

8  quarts  =  1  peck  {ph.), 

4  pecks    =  1  bushel  {iu) 

=  2150.42  cubic  inches. 

The  wine  gallon  of  United  States  '=231        cu.  in. 

The  beer  gallon  of  United  States  =   282        cu.  in. 

The  dry  gallon  of  United  States  =    268.8      cu.  in. 

Imperial  gal.  of  Great  Britain  for  dry 

and  liquid  measures  =    277.274  cu.  in. 

Dry  bushel  of  United  States  =  2150.42    cu.  in. 

Imperial  bushel  of  Great  Britain  =  2218.192  cu.  in. 

TIME    TABLE. 

/     60  seconds  {sec)  =  1  minute  {m,). 

60  minutes  =  1  hour  {hr,). 

24  hours  =  1  day  (da,), 

7  days  =  1  week  (wh). 

30  days  =  1  month  {mo.), 

365  days  =  1  common  year. 

366  days  =  1  leap  year. 


DENOMINATE    NUMBERS.  101 

ANGULAR    OE    CIRCULAR    MEASURE 

is  applied  to  angles  and  circumferences,  reckoning  lati- ' 
tudes  and  longitudes,  etc. 

TABLE. 

60  seconds  {')  =  1  minute  (')• 
60  minutes       =  1  degree  (°). 
30  degrees         =  1  sign  {8.). 
12  signs  or  360  degrees         =  1  circumference. 

Apparently  the  sun  makes  an  entire  revolution  of  the 
earth  in  24  hours,*  and  consequently  travels  15°  in 
1  hour ;  therefore, 

1  hour  of  time  =  15°  longitude. 
1  minute  of  time  =  15'  longitude. 
1  second  of  time  =  15''  longitude. 

MISCELLANEOUS    TABLE. 

12  units  =  1  dozen. 

12  dozen  =  1  gross. 

12  gross  =  1  great  gross. 

20  units  =  1  score. 

24  sheets  of  paper    =  1  quire. 

20  quires  =  1  ream. 

196  lbs.  =  1  barrel  of  flour. 

200  lbs.  ==  1  barrel  of  pork. 

When  a  sheet  of  paper  is  folded  into  two  leaves,  or 
4  pages,  and  a  book  made  in  this  way,  it  is  called  a  folio. 

4  leaves  is  called  a  quarto. 
8  leaves  is  called  an  octavo. 
12  leaves  is  called  a  duodecimo. 
*  BeaUy,  the  revolution  is  that  of  the  earth  on  its  own  axis. 


1()2  DENOMINATE    NUMBERS, 

:>  i'  i  •.*  1»'K  A  CT.I  C  A  L     QUE  ST  IONS. 

1.  Reduce  £3  9s.  lid.  3  far.  to  farthings, 

2.  Reduce  £12  15s.  8d.  to  pence. 

3.  Eeduce  £7  Os.  2d.  to  pence. 

4.  Keduce  2354  farthings  to  the  higher  denominations. 

5.  Reduce  543  pence  to  the  higher  denominations. 

6.  Reduce  731  shillings  to  the  higher  denominations. 

7.  Eeduce  3  T.  6  cwt.  2  qr.  12  lb.  6  oz.  and  9  dr.  to 
drams. 

8.  Reduce  672432  drams  to  the  higher  denominations. 

9.  Reduce  5  lb.  8  oz.  9  pwt.  15  gi\  to  grains. 

10.  Reduce  64324  grains  Troy  to  the  higher  denomi- 
nations. 

11.  Reduce  2  lb.  6  ?  4  3  23  10  gr.  to.  grains. 

12.  Reduce  6742  gr.,  Apothecaries  weight,  to  the 
higher  denominations. 

13.  Reduce  3  lea.  2  mi*  5  fur.  24  rods  2  yd.  1  ft.  6  in, 
to  inches. 

14.  Reduce  802456  inches  to  the  higher  denomina- 
tions. 

15.  Reduce  4  yards  3  qrs.  2  na.  and  2  inches  to  inches. 

16.  Reduce  5  ells  Flemish  to  inches. 

17.  Reduce  4  ells  English  to  inches. 

18.  Reduce  3  ells  French  to  inches. 

19.  Reduce  4  sq.  rods  8  sq.yd.  105  sq.  ft.  and  112  sq.  in. 
to  square  inches. 

20.  Reduce  3  cu.  yd.  12  cu.  ft.  and  1236  cu.  in.  to 
cubic  inches. 

2L  A  pile  of  wood  is  16  ft.  long,  4  ft.  high,  16x4x4 
and  the  length  of  the  wood  is  4  feet.  How  8x4x4 
many  cords  of  wood  ? 


DENOMINATE    NUMBERS.  103 

22.  Eeduce  25  gal.  3  qt.  1  pt.  and  3  gills  to  gills. 

23.  Kediice  9  bu.  3  pk.  4  qt.  1  pt.  to  pints. 

24.  How  many  years,  months,  and  days,  from  April 
15tli,  1842,  to  June  20tli,  1850  ? 


yr,             mo. 
1850            6 
1842            4 

da. 
20 
15 

8            2 

5 

25.  How  many  years,  months,  and  days,  from  October 
25th,  1845,  to  August  18th,  1850? 

yr.              mo. 
1850             8 
1845           10 

da. 
18 
25 

4  9  23 

Rem.— In  this  question,  instead  of  using  the  eighth  and  tenth 
months,  some  authors  prefer  calling  them  7  months  and  9  months ; 
but  if  there  is  an  inaccuracy  in  the  months,  there  is  also  in  the 
years ;  hence  if  we  read  the  above  thus :  The  one  thousand  eight 
hundred  and  fiftieth  year,  the  8th  month  and  25th  day,  there  is 
no  inaccuracy. 

In  computations  of  time  we  always  take  30  days  as  a 
month. 

26.  The  difference  in  time  of  two  places  is  2  hr.  2  min. 
and  2  sec. ;  what  is  the  difference  in  longitude  ? 


hr. 

min. 

sec. 

2 

2 

2 
15 

30^        30'         30" 
Ans.  Thirty  degrees,.  30  minutes,  and  30  seconds. 


104  DENOMINATE    NUMBERS, 

27.  If  the  difference  of  longitude  of  two  places  is  16^^ 
the  difference  of  time  will  be  one  hour;    the  eastern 
place   will  have  the  latest  time.     If  the  difference  in 
longitude  is  16°  24'  30",  what  is  the  difference  in  time? 
15  )  16°  24'  30"  ( 1  hr.  5  min.  38  sec. 

15  10    cts.  =  $yV 

1  12^  cts.  =  li. 

eo  16f  cts.  =  ^. 

-  20    cts.  =  .- 
^^  ^  25    cts.  =  ! 

—  33icts.  =  !„ 
^  37icts.  =  $|. 

>      -^  50    cts.  =  ^, 

570(38  62icts.  =  1 

45__  66f  cts.  =  $|. 

120  75    cts.  =  \ 

120  87icts.  =  \ 

28.  Multiply  576  by  100  =  57600. 

29.  Multiply  576  by  25  =  i  ==  14400.    Take  i  of  the 
above. 

30.  Divide  576  by  100  =  5.76. 

31.  Divide  576  by  25  =  23.04.    Multiply  by  4. 

32.  Multiply  576  by  50  =  ^  (57600)  =  28800. 

33.  Divide  576  by  50  =  5.76  x  2  =  11.52. 

34.  Multiply  576  by  .12^  =  ^  =  $72. 

35.  Multiply  576  by  .16f  =  \  =  $96. 

36.  Multiply  576  by  .33|  =  1  =  8192. 

37.  Multiply  576  by  .62^  =  576  x  5-^8  =  360. 

38.  Multiply  576  by  .87^  =  576  x  7-T-8  —  504. 

39.  .What  cost  342  yds.  muslin  at        8  )_342 

12J  cts.  per  yard  ?  "~^42|=  $42f. 


DENOMINATE    NUMBERS.  105 

40.  Wliat  cost  342  yds.  linen  at  37^  cts.  per  yard  ? 
Multiply  by  3  =  $128^. 

41.  What  cost  342  yds.  linen  at  62^  cts.  per  yard? 
Multiply  by  5  =  $213f . 

42.  What  cost  342  yds.  linen  at  87|^  cts.  per  yard? 
Multiply  by  7  =  $299i. 

43.  What  cost  548  yds.  muslin  at  16f  cts.       6  )_548 
per  yard  ?  ~~$9l-J- 

44.  What  cost  345  yds.  muslin  at  20  cts.       5  )  345 
per  yard  ?  $69' 

45.  What  cost  469  yds.  linen  at  33|  cts.       3  )  469 
per  yard  ?  "IT56-J 

46.  What  cost  469  yds.  linen   at  66|  cts.  per  yard  ? 
Multiply  by  2  ==  $312|. 

47.  What  cost  500  yds.  linen  at  25  cts.  per  yard? 
500  -^  4  =  $125. 

48.  What  cost  500  yds.  linen  at   75  cts.  per  yard? 
Multiply  by  3  =  $375. 

49.  What  cost  500  yds.  linen  at  50  cts.   per  yard? 
500  -^  2  =  $250. 

50.  Bought  648  yards  muslin  at  12^  cts.  a  yard,  and 
sold  it  at  16f  cts.  per  yard.    What  was  the  profit  ? 

■J-  of  648     =     $108 
I  of  648     =        81 

$27/Profit. 

51.  Bought  500  yds.  clolli  at  20  cts.,  and  sold  it  at 
25  cts. ;  what  profit  ? 

i  of  500     ==     $125 
i  of  500     =     _12? 

$25,  Profit. 


106  DENOMINATE    NUMBERS, 

52.  Bought  480  yds.  cloth  at  66-|  cts.,  and  sold,  it  at 
87^  cts. ;  what  was  the  profit  ? 

\  of  480  =  $60;    |  =  $420 
I  of  480  =  160;    |  =   ^ 

$100,  Profit. 

53.  Bought  480  yds.  at  37|-  cts.,  and  sold  it  for  50  cts. 
per  yard  ;  what  was  the  profit  ? 

i  of  480  =  $240 
^  of  480  =  60 ;    |  =  _180 

$60,  Profit. 

54.  Bought  600  yds.  cloth  at  $1  per  yard,  and  sold  it 
for  $1.25  per  yard;  what  was  the  profit  ? 

600   X   li     =     $750 
600   X   1       =       600 

$150,  Profit. 

55.  How  many  yards  of  cloth,  at  12^.  cts.  per  yard,  can 
be  bought  for  $240  ?  8  yds.  can  be  bought  for  every  dollar. 

240  X  8  =  1920  yds. 

56.  How  many  yards  at  16|  cts.  ?  20  cts.  ?  25  cts.  ? 
37i  cts.  ?    50  cts.  ?     62|  cts.  ?     75  cts.  ?     87^  cts.  ? 

37i^  I;    ^^0x1  =  640yds.; 
for  62^  cts.  =.  240  x  f . 

57.  Sold  500  barrels  flour  at  $6.62^  per  barrel,  and 
invested  the  proceeds  in  different  kinds  of  dry  goods, 
averaging  87|^  cts.  per  yard.  What  were  the  proceeds  of 
the  flour,  and  how  many  yards  of  goods  did  I  get  ? 


DENOMINATE  NUMBERS.  107 

58.  How  many  years,  months,  and  days  from  January 
21st,  1874,  to  AprU  30th,  1876  ? 

Ans.  2  y.  3  m.  9  days. 

59.  From  June  18th,- 1865,  to  April  15th,  1869  ? 

Ans.  3y.  9  m.  27  d. 

60.  Prom  Dec.  30th,  1872,  to  Jan'y  5th,  1875  ? 

Ans.  2y.  0  m.  5  d. 

61.  The  difference  in  longitude  of  two  places  is  15°; 
what  is  the  difference  in  time  ?  Ans.  1  hour. 

6^.  The  difference  in  longitude  of  N"ew  York  and  Cin- 
cinnati is  10°  35';  what  is  the  difference  in  time? 

Ans.  42  min.  20  sec. 

63.  The  difference  in  longitude  of  New  York  and  St 
Louis  is  16°  14';  what  is  the  difference  in  time  ? 

Ans.  1  h.  4  min.  56  sec. 

64.  The  difference  in  longitude  of  Philadelphia  and 
Cincinnati  is  9°  20' ;  what  is  the  difference  in  time  ? 

Ans,  37  min.  20  sec. 

65.  The  difference  in  the  time  of  London  and  Wash- 
ington is  5  h.  8  m.  4  sec. ;  what  is  the  difference  in  lon- 
gitude ?  Ans.  77°  1'. 

66.  When  it  is  noon  at  New  York,  what  is  the  time  15° 
east  of  New  York  ?  15°  west  ? 

Ans.  1  P.M.  and  11  A.M. 

67.  When  it  is  noon  at  Cincinnati,  what  is  the  time  at 
New  York  ?  A7is.  12  h.  42  min.  20  sec.  P.M. 

68.  When  it  is  noon  at  New  York,  what  is  the  time  at 
Cincinnati  ?  Ans,  11  h.  17  min.  40  sec.  A.M. 

69.  When  it  is  noon  at  New  York,  what  is  the  time  at 
St.  Louis  ?  Ans,  1  h.  4  min.  56  sec.  P.M. 

70.  When  it  is  noon  at  St.  Louis,  what  is  the  time  at 
New  York  ?  Ans.  10  h.  55  min.  4  sec.  A.M. 


108  DENOMINATE  NUMBERS. 

71.  When  it  is  noon  at  Philadelpliia,  what  is  the  time 
at  Cincinnati?  Ayis.  11  h.  22  min.  40  sec.  A.M. 

72.  When  it  is  noon  at  Cincinnati,  what  is  the  time  at 
Philadelphia  ?  Ans.  12  h.  37  min.  20  sec.  P.M. 

73.  When  it  is  noon  at  Washington,  what  is  the  time  at 
London  ?  Ans,  5  h.  8  min.  4  sec.  P.M. 

74.  Reduce  4  shil.  10  pence  and  2  far.  to  the  decimal 
of  a  <£.  Ans.  £.24375. 

Cor.  Reduce  the  compound  number  to  the  lowest  given 
denomination,  which  will  be  the  numerator  of  a  common 
fraction,  the  denominator  of  which  must  be  the  number 
of  the  lowest  denomination  that  makes  a  unit  of  the  one 
to  which  it  is  to  be  reduced ;  then  perform  the  division 
indicated  by  the  fraction,  adding  decimal  zeros  to  the 
numerator. 

Rem.  In  general,  I  prefer  the  common  fraction  to  the 
decimal. 

75.  Express  £5  12s.  4d.  3  far.  in  pounds  and  decimals. 

Ans.  £5.61979f 

76.  Reduce  the  qrs.  and  lbs.  to  the  decimal  of  cwi;  6 
cwt.  2  qrs.  10  lbs.  Ans.  6.6  cwt. 

77.  Reduce  57  lbs.  to  the  decimal  of  a  cwt. 

Ans.  .57  cwt. 
Rem.  Pounds  are  as  readily  reduced  to  the  decimal  of 
cwt.  as  pounxis  are  hundredths. 

78.  Reduce  £12  9s.  4d.  3  far.  to  farthings. 

Ans.  11,971  far. 

79.  Reduce  £27  14s.  8d.  1  far.  to  farthings. 

Ans.  26,625  far. 

80.  Reduce  £8  12s.  9d.  to  pence.     '        Ans.  2,073d. 

81.  Reduce  £9  8s.  to  shillings.  Ans.  188s. 


DENOMINATE  NUMBERS.  109 

82.  Eeduce  24,862  far.  to  the  higher  denominations. 

Ans.  £25  17s,  lid.  2  far. 

83.  Reduce  3,684d.  to  the  higher  denominations. 

Ans.  £15  7s. 

84.  Add  £3  10s.  8d.  2  far.,  £12  7s.  8d.  3  far.,  and  £7 
4s.  2d.  Ans.  £23  2s.  7d.  1  far. 

85.  From  £12  8s.  4d.  1  far.  take  £7  9s.  5d.  3  far. 

Ans.  £4  18s.  lOd.  2  far. 

86.  Multiply  £3  5s.  7d.  3  far.  by  5. 

A71S.  £16  8s.  2d.  3  far. 

87.  Multiply  £8  6s.  8d.  2  far.  by  12. 

Ans.  £100  Os.  6d. 

88.  Divide  £24  8s.  9d.  by  12.    Ans.  £2  Os.  8d.  3  far. 

89.  Divide  £46  8s.  6d.  by  9.  Ans.  £5  3s.  2d. 

90.  Eeduce  12s.  8d.  2  far.  to  the  fraction  of  a  £. 

Ans.  f|. 

91.  Eeduce  7s.  3d.  to  the  decimal  of  a  £.       Ans.  ff^. 

92.  Eeduce  2  qrs.  15  lbs.  to  the  fraction  and  to  the 
decimal  of  a  cwt.  Ans.  ^  and  .65. 

93.  Eeduce  12  oz.  avoirdupois  to  the  fraction  and  to 
the  decimal  of  a  cwt.  Ans.  -^^-^  and  .0075. 

94.  Eeduce  8  oz.  Troy  to  the  fraction  and  to  the  deci- 
mal of  a  lb.  Ans.  |  and  .666f. 

95.  12  cents  is  what  part  of  one  dollar  ? 

Ans.  $^or$.12. 

96.  12  lbs.  is  what  part  of  a  cwt.? 

Ans.  -^  cwt.  or  .12  cwt. 

97.  3  qrs.  and  7  lbs.  are  what  part  of  a  cwt.  ? 

Ans.  fj-  cwt.  or  .82  cwt. 

98.  If  chestnuts  are  worth  11.60  a  bushel,  what  is  a 
quart  worth  ?  Ans.  h  cents. 


110  DENOMINATE  NUMBERS. 

99.  How  many  pint  bottles  can  be  filled  from  31|  gaL 
cider?  Arts.  252  bottles. 

100.  What  is  the  value  of  a  pile  of  wood  32  ft.  long,  12 
feet  high,  and  4  feet  wide,  at  $5  a  cord  ?        Ans,  $60. 

101.  If  a  man  earn  $9.25  a  week,  how  much  will  he 
earn  in  5  weeks  ?  Arts,  $46.25. 

102.  A  man  earned  $95.37^,  and  spent  $34.87^ ;  how 
much  had  he  left  ?  Ans,  $60.50. 

103.  Bought  37f  yards  of  cloth  at  $5.25  per  yard ;  what 
was  the  cost  ?  Ans.  $198.18|-. 

104.  What  is  the  cost  of  257|  acres  of  land  at  $37.45  per 
acre?  Ans.  $9,652. 73f. 

105.  What  is  the  cost  of  542:^  bushels  of  wheat  at 
$1.12^  per  bushel?  Ans.  $610. 03|. 

106.  What  is  the  product  of  ^  and  .625  ? 

■^ns.  ^,  or  .0925. 

107.  What  part  of  5  gals.  3  qts.  1  pt.  is  3  gal.  2  qt.  1  pt.? 

Ans,  1^. 
Def.  The  reciprocal  of  a  number  is  formed  by  invert- 
ing the  number  thus:   2,  3,  4  may  be  written  f ,  |,  \', 
their  reciprocals  are  |,  |,  ^;   so  also  of  any  fraction — 
f,  f,  I,  the  reciprocals  |,  -f,  |. 

108.  What  is  the  value  of  any  number  multiplied  by  its 
reciprocal?  Ans,  1. 

109.  How  much  alloy  must  be  mixed  with  2  lb.  8  oz. 
pure  gold  to  make  it  18  carats  fine  ? 

Pure  gold  is  24  carats  fine.    Ans.  10  oz.  13  pwt.  8  gr. 

110.  If  4  equal  angles  are  made  at  the  center  of  a  circle, 
how  many  degrees  in  each  ?  Ans,  90  degrees. 

These  angles  are  termed  right  angles. 


37- 


Ratio. 


Two  fractions  can  be  formed  with  any  two  integral 
numbers,  the  one  a  proper  fraction  and  the  other  an 
improper  fraction ;  thus,  ^  and  \  can  be  formed  with  7 
and  9.  When  the  proper  fraction  is  a  multiplier  of  any 
number,  the  product  is  less  than  the  number  multiplied; 
therefore,  this  fraction  is  termed  a  JDifninishing 
Ratio,  But  when  the  improper  fraction  is  a  multiplier, 
the  product  is  greater  than  the  multiplicand;  hence,  the 
Improper  fraction  is  termed  an  Increasing  Ratio. 

PROBLEMS. 

1.  If  5  lbs.  sugar  cost  50  cents,  what  will  9  lbs.  cost  ? 

It  is  evident  that  9  lbs.  will  cost  more  than  5,  and  just 
as  much  more  as  is  indicated  by  the  increasing  ratio 
formed  by  the  two  like  terms,  5  lbs.  and  9  lbs. 

10 

If  5  lbs.  cost  50  cts.,  9  lbs.  will  cost  ^0  cts.  x  f  =  90  cts. ; 
this  may  be  further  demonstrated  thus, 

5  lbs.  =  50  cts. 
1  lb.  =  10  cts. 
9  lbs.  =  90  cts. 

In  a  problem  of  ratios,  the  one  ratio  is  given,  and  one 
of  the  terms  of  the  other  ratio,  to  get  the  second  term ; 
thus,  in  the  above : 


112  RATIO. 

Given,  5  lbs.  sugar    and    50  cents. 

Kequired,       9  ibs.  sugar    and         ? 

The  ratio  of  the  money  will  be  the  same  as  of  the  sugar. 
As  the  required  sugar  is  more  than  the  given,  the  ratio 
must  be  increasing ;  that  is,  |.  .*.  50  x  -^  =  90,  the  ratio 
of  the  required  money  to  the  given,  \^=.\,  the  same  as 
of  the  sugar. 

EXAMPLES. 

1.  If  5  bushels  of  wheat  cost  $6.25,  what  will  8  bushels 
cost? 

Given  5  bu.    and    $6.25. 

'Eequired,      8  bu.    and 

%^.U  X I  =  $10.00. 

8  __25)1000  __  5)40  __  8 
5  ""  625   ~"       25  ~  5* 

The  ratios  of  the  wheat  and  of  the  money  is  the  same. 
Rem. — Ratios  can  only  be  formed  by  two  like  terms. 

2.  If  5  bushels  of  oats  cost  $1.50,  what  will  21  bushels 
cost? 

^  z=i  fll  =  ^;  the  ratio  is  the  same. 

Given  5  bu.    and    $1.50. 

Eequired,      21  bu.    and 

$1.^0  X  -^  =  $6.30. 

Rem. — ^Write  the  given  terms  in  a  line  and  the  like  term  of  the 
required  immediately  under  that  of  the  given.  One  term  of  the 
required  is  wanting,  and  the  given  like  term  may  be  called  the 
term  of  demand,  and  should  be  placed  first  and  multiplied  by  the 
ratio,  having  for  its  numerator  the  required  term  of  the  ratio,  and 
for  its  denominator  the  given  term. 


RATIO,  113 

As  a  general  thing,  an  increase  in  the  required  term  of  the  ratio 
will  take  more  of  the  unknown  to  accomplish  it ;  an  increased 
amount  of  goods  will  cost  a  greater  sum  ef  money ;  an  enlarged 
piece  of  work,  an  additional  sum  of  money ;  and  the  greater  the 
work,  the  longer  time  to  perform  it,  etc.  In  examples  of  this 
kind,  the  ratios  are  direct,  and  the  required  term  of  the  ratio  holds 
the  place  of  the  numerator  and  the  given  term  that  of  the  denomi- 
nator, and  the  product  of  the  ratio  and  the  odd  given  term  is  the 
term  required. 

3.  If  a  man  travel  40  miles  in  8  hours,  how  many  miles 
will  he  travel  at  that  rate  in  18  hours  ? 

Given  40  miles    and    8  hours. 

Eequired,      ?    miles    and  18  hours. 

6 

^0  miles  X  J^  ==  90  miles. 

4.  If  15  bushels  of  wheat  yield  3  barrels  of  flour,  how 
many  bushels  will  yield  10  barrels  of  flour? 

Given  15  bu.    and      3  barrels  of  flour. 

Kequired,         ?  and    10  barrels  of  flour. 

SOLUTION. 
6 

-10  X  -^  =  50  bushels. 

5.  If  a  man  travel  30  miles  in  2  days,  how  long  will  it 
take  him  to  travel  240  miles  ? 

Given  30  miles    and    2  days. 

Eequired,    240  miles    and        ? 

6.  If  a  staff  4  feet  long  cast  a  shadow  3  feet,  what  is 
the  height  of  a  steeple  which  casts  a  shadow  90  feet? 

STAFF.  SHADOW. 

Given  4  ft.  3  ft. 

Eequired,        ?  90  ft. 


114  RATIO, 

7.  If  the  interest  of  $100  for  one  year  is  $5,  what 
would  be  the  interest  of  $500  for  the  same  time  ? 


PRmCIPAL. 

INTEREI 

Given 

$100 

and 

$5. 

Required, 

"$500 

and 

V 

8.  If  f  of  a  barrel  of  flour  cost  $4,  what  will  4|  barrels 
cost  ? 

Given  f  barrel    and    $4. 

Eequired     4f  barrel    and     ? 


7 


Rem.  4|  is  a  multiplier,  and  f  is  a  divisor ;  the  |  must  be 
inverted. 

9.  If  4|  bushels  of  wheat  cost  $5.40,  what  will  8|  bu. 

cost?  9ibu.?  23i?  31f?  47^?  39^?  58f?  97^?  106|? 

10.  If  8  bushels  of  wheat  cost  $10,  what  will  be  the 

cost  of  ^  bu.  ?  lOi  bu.  ?  15|  bu.  ?  37^  bu.  ?  95f  bu.  ? 

125f  bu.  ?  150|  bu.  ?  279f  bu.  ? 

,^  11.  If  5|  acres  of  land  cost  $280,  what  is  the  cost  of 
6i  acres?  7|  acres?  12^  acres?  13^  acres  ?  17|?18|? 
19|?  20|?  37|?  49|? 

12.  If  2\  acres  of  land  cost  $110,  what  will  |  of  an 
acre  cost?  |acre?  J  acre?  \  acre?  |  acre?  -f  acre? 
f  acre  ?  1-^  acres  ?  1 1  acres  ? 

13.  If  -^7^  of  a  yard  of  cloth  cost  y^^^  of  a  dollar,  what 
will  "I  of  a  yard  cost  ? 

Given  -^^  yd.     and    %^. 

Eequired,       f  yd.     and      ? 
tV  X  f  X  ^. 


BATIO.  115 

14.  If  I  yard  of  cloth  cost  $2,  what  will  3  ells  Fl.  cost? 

2  x|  = 

15.  If  f  yard  of  cloth  cost  $2.25,  what  will  5  ells 
English  cost  ? 

2.25  X  ^. 

What  will  5  ells  French  cost  ? 

In  the  preceding  examples,  the  ratios  were  all  direct; 
as  in  those  cases  any  increase  in  the  required  term  of  the 
ratio  demanded  a  similar  increase  of  the  unknown ;  but  . 
there  are  cases  which  require  the  ratio  to  be  inverted, 
such  as,  the  more  men  employed,  the  less  time  will  be 
required  to  perform  a  piece  of  work;  the  more  hours 
employed  in  the  day,  the  less  days ;  the  wider  the  ma- 
terial, the  less  yards  it  will  take  to  make  a  garment,  etc. 

These  cases  of  inverse  ratio  are  readily  detected  by 
asking  this  question :  "  Will  an  increase  of  the  required 
term  of  the  ratio  demand  an  increase  in  the  unknown 
term  ?"  If  it  does,  the  ratio  is  direct;  but  if  an  increase 
in  the  required  term  of  the  ratio  demand  a  diminution 
of  the  unknown  term,  the  ratio  must  be  inverted ;  thus, 

PROBLEM. 

If  4  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  8  men  to  do  the  same  work  ? 

Given  4  men  in  10  days. 

Required,      8  men  in        ? 

5 

^0  days  X  I  =:  5  days. 


116  RATIO. 

Cor.— It  is  evident  that  8  men  will  do  it  in  less  time  ; 
that  is,  in  one-half  the  time  that  it  will  take  4,  which 
ratio  is  expressed  by  the  diminishing  ratio  of  4  and  8, 
that  is,  I  =  1^,  in  which  the  given  term  is  the  numerator 
of  the  ratio  and  the  required  term  the  denominator. 


EXAMPLES. 

1.  If  5  men  can  dig  a  ditch  in  20  days,  how  many  men 
will  dig  it  in  25  days  ? 

Given  5  men    and    20  days. 

Eequired,  ?        and    25  days. 

4 

$  men  x  |f  =  4  men. 

Ik 

Hem. — An  increase  in  the  d&js  will  require  less  men. 

2.  If  6  horses  eat  a  certain  quantity  of  hay  in  30  weeks, 
how  many  horses  will  consume  the  same  quantity  of  hay 
in  9  weeks  ? 

Given  6  horses    and    30  weeks. 

Eequired,  ?         and      9  weeks. 

2  10 

0  horses  x  ^  =  20  horses. 


3.  K  a  man  perform  a  journey  in  12  days,  when  the 
days  are  9  hours  long,  how  many  days  of  12  hours  will  it 
take  him  ? 

Given  12  days    and    9  hours. 

Eequired,  ?  and  12  hours. 


RATIO.  117 

4.  How  many  yards  of  lining  |  yd.  wide  will  it  take  to 
line  3  yards  of  cloth  f  yd.  wide  ? 

Given  f  yard  wide    and    3  yards  long. 

Eequired,      f  yard  wide    and  ? 

$  X  I  =  6  yds. 

COMPOUND    KATIO. 

When  there  are  two  or  more  ratios,  it  is  termed 
Compound  Ratio  ;  thus, 

If  3  men  in  12  days  build  40  rods  of  wall,  how  many 
rods  will  9  men  build  in  24  days  ? 

Given  3  men,    12  days,    40  rods. 

9  men,    24  days,         ? 

40  X  I  X  fl  =:  240  rods. 

4 

Rem. — Each  ratio  is  direct. 

If  12  men  dig  a  ditch  20  rods  long  in  18  days  by  work- 
ing 8  hours  a  day,  how  many  men  will  dig  a  ditch  40  rods 
long  in  24  days,  working  6  hours  a  day  ? 

Given  12  men,    20  rods,    18  days,    8  hours. 

?  40  rods,    24  days,    6  hours. 

2  ^ 

12  men  x||xi|x|  =  24  men. 

Exemplification.— The  longer  the  trench,  the  more  men  it 
will  take,  and  the  ratio  is  direct ;  but  the  greater  the  number  of 
days  and  the  more  hours  of  each  day,  the  less  men  would  be  re- 
quired ;  hence  these  two  ratios  are  inverse. 

Cor. — Each  ratio  must  be  dealt  with  as  in  the  pre- 
ceding article. 


118  RATIO, 


EXAMPLES. 

1.  If  3  men  in  8  days  of  9  hours  each  build  a  wall  20  ft. 
long,  2  ft.  thick,  and  4  ft.  high,  in  how  many  days  of 
8  hours  each  will  12  men  build  a  wall  100  ft.  long,  3  ft. 
wide,  and  6  ft.  high  ? 

Given     3  men,  8  days,  9  hrs.,    20  ft.  1.,  2  ft.  w.,  4  ft.  h. 
12  men,        ?      8  hrs.,  100  ft.  1.,  3  ft.  w.,  6  ft.  h. 

$  days  X  A  X  I  X  -W  X  I  X  }  =  ^^^^^/^ 
=  W  =  ^^^  days. 
Bem. — The  men  and  hours  are  inverse,  the  other  ratios  direct. 

>  2.  If  6  men  mow  12  acres  of  grass  in  2  days  of  10  hrs. 
each,  how  many  hours  a  day  must  8  men  work  to  mow 
40  acres  in  4  days  ?  Ans,  12|^  hours  a  day. 

I-  3.  If  6  horses  eat  36  bushels  of  oats  in  18  days,  how 
many  bushels  will  be  sufficient  for  12  horses  24  days  ? 

A71S.  96  bushels. 
4.  If  $100  in  12  months  gain  $6,  how  long  will  it  take 
$500  to  gain  $15  ? 


Given 
Eequired, 

$100 
$500 

prin., 
prin.. 

12  mo., 

? 

$6  int. 
$15  int. 

l^mo, 

.xi 

x4  = 

=  6  months. 

5.  If  6  men  manufacture  300  pairs  of  shoes  in  30  days, 
how  many  men  will  make  900  pairs  of  shoes  in  60  days  ? 

Ans,  9  men. 


Percentage. 


Pev  Cent,  means  per  hundred,  and  is  generally  ex- 
pressed fractionally;  thus,  5  per  cent.,  6  per  cent., 
marked  b%  and  6^,  is  expressed  y^,  yg^,  etc.,  or  .05,  .06 ; 
thus,  yfo  of  100  =  100  X  y|^  =  5,  and  -^  of  100  is  6. 

EXAMPLES. 

1.  What  is  b%  of  200  ?  ^00  x  ,1^  =  10,  Ans. 
What  is  5^  of  300  ?  Arts.  15. 
What  is  b%  of  400  ?  Ans.  20. 

2.  What  is  5^  of  245  ?         2.45  X  yf^  =  12.25,  Ans. 
The  100  is  canceled  in  the  245  by  pointing  off  two 

places  of  decimals.  3 

3.  What  is  %%  of  300  ?  $00  x  li^  =  18,  Ans. 
What  is  6^  of  400  ?  Ans.  24. 
What  is  %%  of  500  ?  Ans.  30. 

4.  What  is  6^  of  368  ?  3.68  X  yf^  =  22.08. 

COMMISSIOlSr,    OR    BEOKEKAGE. 

The  business  of  a  commission  merchant  or  broker  is  to 
make  purchases  and  sales,  on  which  he  receives  a  per- 
centage. 

PROBLEM    I. 

A  purchase  of  $100  worth  of  goods,  at  1%  commission, 
will  cost  $101 ;  that  is,  \^  of  the  amount  of  the  purchase. 


120  PERCENTAGE, 

PROBLEM     II. 

In  a  sale  of  goods  for  $100,  at  1%  commission,  the 
owner  will  realize  $99  ;  that  is,  -^  of  the  amount  of  sale. 

'PROBLEM    III. 

When  stocks,  bonds,  drafts,  or  currency,  are  purchased 
at  a  discount  of  2%,  the  cost  of  $100  worth  will  be  $98; 
that  is,  -^  of  the  face  of  the  bond,  etc. ;  but  when  they 
are  purchased  at  a  premium  of  2^,  the  cost  of  $100  worth 
is  $102 ;  that  is,  \^  of  the  face. 

PROBLEM    IV. 

In  the  exchange  of  currency,  when  there  is  a  premium 
on  the  funds  on  hand,  as  that  of  English  money  to  be 
exchanged  into  United  States,  the  premium  in  favor  of 
England  is  about  9^ ;  it  is  computed  as  follows : 

Eng.  £  X  ^^  X  i^  =  $  U.  S., 

and  $  TJ.  S.  X  A  X  iU  =  ^  Eng. 

that  is,  England  gets  $109  for  every  $100  of  her  money, 
and  the  United  States  must  pay  $109  of  her  money  for 
$100  English  money. 

Rem. — This  is  according  to  the  old  exchange  value.  Now, 
however,  the  exchange  value  of  £1  is  fixed  at  $4.86. 

EXAMPLES. 

1.  A  broker  sold  goods  to  the  amount  of  $6000,  at  2% 
commission,  and  invested  the  balance  of  the  proceeds, 
after  deducting  2%  on  the  amount  of  purchase ;  what 
was  the  owner's  portion  of  the  sale,  and  what  amount  of 
goods  were  purchased  ? 


PERCENT  A  GE.  121 

$6000  Xt^  =  $5880  —  owner's  portion  of  the  sale. 
$5880  x|if  =  $5764.7011  ^  amt.  of  goods  purchased. 

.-.  6000x4\xMI  =  6000xif 

51 

=  $576441-,  amt.  of  goods  purchased. 

Rem. — Observe  the  difference  in  the  ratios  of  the  sale  and 
purchase. 

^.  What  is  the  cost  of  a  bond  for  $5000  at  h%  discount, 
stocks  whose  face  indicate  $2000  at  \.%  premium,  $1000 
currency  at  2^  discount,  and  $3000  gold  at  8^  premium  ? 
$5000  X  ^^     =  $4750 

2000  X  itt     =  2080 

1000  X  ^     =  980 

3000  X  HI     =  3240 

$11050 
Cor. — When  brokerage  is  paid  in  the  exchange  of 
money,  the  percentage  is  on  the  amt.  purchased,  which, 
if  the  rate  is  2%,  is  |f|  of  the  funds  on  hand. 

3.  If  a  broker  makes  sales  to  the  amount  of  $500,  on 
which  he  receives  S%,  what  is  his  commission  ? 

5 

2.  What  is  the  cost  of  a  draft  for  $1000  at  a  premium 

$1000  X  «  X  ^^^  =^  $1005. 

3.  What  is  the  face  of  a  draft  at  i%  premium,  costing 
11005  ? 


122  PERCENTAGE, 

4.  A  broker  makes  sales  for  $4325,  at  2^ ;  what  is  the 
brokerage,  and  what  does  the  owner  reahze  ? 

43.25  X  xt^  =  $86.50,  commission. 
43.25  X  ^^  =  $4238.50,  owner  reahzed. 

5.  A  merchant  sells  to  a  broker  $3275  uncurrent  funds 
at  5%  discount ;  what  does  he  realize  ? 

163.75  X  19 
$$M-$x  ^\\  =  $3111.25. 

6.  An  architect  charges  1^%  for  plans  and  specifica- 
tions, and  2^%  for  superintending  a  building,  the  cost  of 

.which  is  $10000  ;  what  is  the  architect's  fees  ?  Ans,  $400. 

7.  A  broker  has  2%  commission,  and  3%  for  guarantee- 
ing payment ;  what  does  he  receive  on  sales  amounting 
to  $42325?  A71S.  $2116.25. 

8.  I  sent  my  broker  $4000  to  purchase  goods;  what 
amount  of  goods  did  he  purchase  after  deducting  com- 
missions at  2%  on  the  amount  of  goods?    $4000  x  fgf. 

For  every  $102  he  gets  $100  worth  of  goods. 

9.  Bought  a  draft  on  New  York,  the  face  of  it  $500 
premium,  i%  ;  what  is  the  whole  cost  and  the  premium  ? 

«»  -  S  X  ^  =     *^-^ 

500.00 


Premium,      $1.25 

10.  Sold  goods  to  the  amount  of  $4444,  and  invested 
the  proceeds,  after  retaining  my  commissions,  which 
were  2%  on  the  sales,  and  1%  on  the  investment;  what 
was  the  amt.  of  investment  ? 

•      $MM  X  ^Ax^  =  $4312. 


PERCENTAGE.  123 


INTEREST. 

Interest  is  an  allowance  for  the  use  of  money.  It  is 
reckoned  by  percentage;  thus,  b%,  6%,  etc.,  meaning  for 
a  year,  when  not  otherwise  expressed;  for  any  other 
time  it  is  as  the  ratio  of  the  time ;  thus,  the  interest  of 
$100  at  6%  is  $6  for  a  year,  for  two  years  $12,  and  for 
six  months  $3. 

PROBLEMS. 

1.  Find  the  interest  of  $150,  at  6%,  for  1  year, 

$150  X  tU  =:=  $d. 
For  8  months. 

$1.50  X  tI^  X  A  =  1.50  X  ^  =  $6.00. 

For  6  months. 

$1.50  X  ^  X  A  =  $4.50. 

For  14  months. 

7 

$1.50  X  tI^  X  M  =  1-50  X  xfe  =  110.50. 

Cor. — At  6%,  the  rate  per  cent,  for  any  number  of 
months  is  ^  the  number  of  months;  thus,  for  8  montb« 
it  is  4^,  for  6  months  it  is  3%,  and  for  14  months  7%. 

2.  Find  the  interest  of  $150,  at  6%,  for  129  days  ? 
$150  X  T^  X  ilt  =  1150  X  i^  =  $3,225. 

60 

CoR. — The  interest  of  a  sum  of  money  for  any  number 
of  days  is  equal  to  the  product  of  the  sum  of  money  and 


124  PERCENTAGE. 

the  number  of  days  divided  by  6000;  or,  if  the  number 
of  dollars  be  multiplied  by  the  number  of  days  and  this 
product  divided  by  6,  the  quotient  is  the  interest  in 
mills ;  point  off  three  decimals  and  it  is  reduced  to  dol- 
lars, cents,  and  mills. 

If  the  rate  of  interest  is  7^,  add  \ ;  if  8^,  add  -J- ;  if  9^ 
*^add  ^ ;  if  5^,  deduct  -J-;  if  4^,  deduct  ^ ;  if  3^,  take  ^. 

The  rate  for  200  months  is  100^ ;  that  is,  the  interest 
is  equal  to  the  principal. 

200  months  of  $100  is  $100. 

20  months  of  $100  is  $10. 

2  months  of  $100  is  $1. 
30  days,  or  1  month  $0.50. 

3  days  of  $100  is  $.05. 

1  day  of  $100  is  $.01f . 

2  days  of  $100  is  $.03-^. 

EXAM  PLES. 

1.  Find  the  interest  of  $625,  at  6^,  for  8  months. 
Rate  for  8  mo.  is  y^. 

$6.25  X  T*^  =  $25.00. 
For  8  mo.  and  20  days,  8|  mo.,  rate  4t\%. 

$625  X  ^^     =  ml  X  ^  =  $27.08^. 

If  months  and  days  are  computed  separately. 

$625  X  tU  =  $25.0* 

$625  X  ^h  =  _^3 

$27. 08^ 


PER  CENT  A  GE,  125 

2.  What  is  the  interest  of  $650,  at  6^,  for  1  year 
6  months  and  24  days  ?    At  7^  ?    At  %%  ? 
Eeduce  to  days  1  year     =     360 
Eeduce  to  days  |  year    =     180 

24 
564 


$650 

X  'KB.U^  — 

$61.10  =  int.  at 

6^. 

1  000 

iofl8||  = 

H^%  =  9, 

.4^. 

$650 

X  loJo  — 

$61.10  =  int.  at 

e%. 

At  6^  .= 

:      $61.10 

At  6^    = 

$61.10 

Addi    = 

:         10.18i 

Addi    = 

20.361 

Mt%    = 

:     $71.28^ 

At  8^    = 

$81.46| 

3.  What  is 

the  interest  of  $575,  at  6%, 

for  2  years 

4  months  and  18  days  ? 

2  years 

= 

720  days. 

4  months 

120  days. 
18  days. 

858  days. 

143 

$575  X 

^  =  575  X 

^-^  =  $82,225. 

1000 

-V-  4.  A  note  due  April  1st,  1872,  for  $2000,  bearing 
Interest  at  6%.  On  the  back  of  the  note  were  the  follow- 
ing credits:  May  1st,  1873,  $230;  June  1st,  1874, 
$223.50 ;  July  1st,  1875,  $217.  What  was  due  July  1st, 
1877,  when  the  note  was  taken  up  ?  Ans.  $1904. 

First  credit        $230. 

Second  credit     $223.50. 

Third  credit      $217.      v  j 


126  PER  CENTA  G^. 

5.  A  note  dated  1st  April,  1870,  for  $200,  bearing 
interest  from  date  at  Q%,  has  the  following  endorsements 
on  the  back  of  it. 

May  1st,  1871,  paid  812. 
May  1st,  1872,  paid  112. 
May  1st,  1873,  paid  $12. 

What  was  due  on  the  note  April  1st,  1875,  when  it 
was  paid  ? 

Int.  of  $200  for  1  year  is         $12 

5  years. 

$60 
Add  Principal,  200 

$260 
Deduct  sum  of  payments,        36 

April  1st,  1875,  balance,       $224  paid. 

As  the  sum  of  the  payments  never  equaled  the  interest, 
no  computation  need  be  made  until  the  end,  when  the 
sum  of  the  payments  must  be  deducted. 

Pkob.  1. — In  the  case  of  Partial  Payments,  when  the 
payment  is  greater  than  the  interest  until  the  time  of 
the  payment,  the  interest  is  computed  until  the  date  of 
the  payment  and  added  to  the  principal,  and  from  this 
sum  the  payment  is  deducted,  and  the  balance  is  regarded 
as  the  principal  of  the  note. 

Peob.  2. — When  the  payment  is  less  than  the  interest, 
no  computation  is  made ;  but  whenever  the  sum  of  the 
payments  is  greater  than  the  interest  until  that  time, 
then  the  interest  is  to  be  computed  to  the  date  and  added 
to  the  principal,  and  from  this  sum  the  sum  of  all  the 


PER  CENT  A  GE.  127 

payments  is  deducted,  and  the  balance  regarded  as  the 
principal  of  the  note. 

These  computations  are  to  be  repeated  until  the  note 
is  paid, 

BANK  DISCOUNT. 

Sanh  Discount  is  reckoned  the  same  as  interest 
on  the  face  of  the  note  and  for  the  time  the  note  is  given, 
plus  three  days,  which  are  called  days  of  grace,  and  the 
note  need  not  be  paid  until  the  last  day  of  grace,  three 
days  after  the  time  specified  in  the  note. 

It  is  called  Discount,  because  the  borrower  does  not 
receive  the  sum  specified  in  the  note,  but  the  difference 
of  this  sum  and  the  interest. 

Notes  in  Banks  are  usually  given  for  a  short  time,  viz., 
for  30  days,  60  days,  or  90  days;  and  the  interest  is  com- 
puted for  33  days,  63  days,  or  93  days. 

1.  The  bank  discount  on  a  note  for  $100  at  30  days. 

$100  X  ^  X  ^%  =  ii  =  $.55. 

60 

Borrower  gets  $100  —  $.55  =  $99.45. 

2.  Borrowergets$100— $1.05  =  $98.95.   $100  for  60  da. 

W00  X  tIis  X  ^«A  =  li  =  $1.05. 

60 

3.  Borrower  gets  $100— 1.55  rr  $98.45.   $100  for  90  days. 

$100  x^x  i^  =  ^  =  $1.55. 

60 

4.  Borrower  gets  $324— $5.02  =:  $318.98.  $324  for 
90  days.^ 

'^  16  2  31 

UU  X  ^11^  =  $5,022.     ^ 


128  PERCENTAGE, 


TRUE    DISCOUISTT 

is  the  abatement  made  on  a  note  not  yet  due,  or  the 
difference  between  the  face  of  the  note  and  a  sum  of 
money  which,  placed  at  interest,'  will  amount  to  the 
face  of  the  note  by  the  time  the  note  is  due ;  thus,  the 
present  value  of  a  note  for  $100,  due  in  one  year  without 
interest,  when  money  is  worth  6^,  is  such  a  sum  as  will 
amount  to  $100  in  one  year  at  Q%  interest. 

CoR.— As  $100  now  will  be  worth  $106  at  the  end  of 
the  year,  so  the  present  worth  of  money  due  in  one  year 
is  \l^  of  the  face  of  the  note  due  in  one  year. 

CoR.— The  ratio  to  obtain  the  present  worth  from 
the  face  of  the  note,  has  for  its  numerator  100,  and 
the  denominator  is  100  increased  by  the  interest  for 
the  time  and  rate. 

EXAMPLES. 

1.  What  is  the  present  worth  of  a  note  for  $324,  du© 
in  one  year,  rate  of  money  Q%  ? 

$324  X  iif  =  $305.66  +  ,  present  worth. 
$18.34,  discount. 

2.  Due  in  2  years  1  month. 

100 


Bate  = 

9  months  = 


100 
104i 


200 
225  " 

8 
"  9 

200 

209 

PER  CENT  A  GE,  129 


EXCHANGE. 

Def.  1. — Exchange  is  the  system  by  which  pay- 
ments are  made  at  a  distant  place  by  means  of  Bills  of 
Exchange  or  Drafts. 

2.  The  person  making  or  signing  the  Bill  or  Draft,  is 
called  the  Maker  or  Drawer  ;  the  one  to  whom  it  is 
addressed  is  the  Drawee^  and  the  person  to  whom  it  is 
ordered  to  be  paid  is  the  Payee* 

3.  The  person  in  possession  of  the  draft  is  the 
Molder,  and  if  he  endorse  it,  he  becomes  responsible 
for  the  payment,  unless  otherwise  specified. 

4.  Exchange  between  the  different  cities  of  one's  own 
country  is  Domestic^  and  that  with  a  foreign  country 
is  Foreign  Exchange. 


DOMESTIC    EXCHANGE. 

PROBLEM. 

To  find  the  cost  of  a  draft  on  Philadelphia  or  New 
York  when  at  a  premium,  and  also  when  at  a  discount ; 
thus,  if  the  premium  is  1%,  the  draft  will  cost  ^^ 
of  its  face  ;  if  at  a  discount  of  1%,  it  will  cost:  -^  of 
its  face. 

Rem. — The  face  of  the  draft  must  be  equal  to  the  sum  of 
money  tliat  we  wish  to  remit. 


130  PER  CENT  A  GE. 


EXAM  PLES. 


1.  What  is  the  cost  of  a  draft  on  Philadelphia  for  $500, 
at  \%  premium  ? 

500  X  ^  =  500  X  IM  =  ^^¥^  =  502i. 

2.  I  owe  $3000  in  New  York,  and  the  premium  on  a 
New  York  draft  is  \%.    What  must  I  pay  for  the  draft  ? 

$$000  x^  =  tM  =  lSx^=:  $3007.50. 

3.  What  is  the  cost  of  a  draft  on  New  Orleans  for 
$500,  at  1^  discount? 

500  X  II  =  497i. 

Rem. — The  computation  may  be  made  as  interest ;  thus,  first 
example,  $500  at  1  %  is  $5,  and  \  of  $5  is  $3J  =  premium ;  second 
example,  $3000  at  1^  is  $30,  at  J%  it  is  $7.50  premium ;  in  the 
third  example,  $500  at  J%  is  $2.50,  the  discount. 

5.  What  is  the  cost  of  a  draft,  at  60  days,  on  New 
York  for  $500,  premium  \%y  interest  off  at  6%  ? 

Discount     =     %%%     =    U%. 
Premium     =      i%     =    l%%. 

Discount  above  premium    =     ^%. 
X  ^^  =  $2.75,  discount. 

4 

$500 

2.75 


$497.25,  cost  of  draft. 


t 


i^'- 


PERCENTAGE.  131 


FOREIGN   EXCHANGE. 

EXAMPLES. 

1.  What  is  the  cost  in  New  York  of  a  draft  on  London 
for  £500,  exchange  at  $4.80  to  the  £  ? 

£500  X  4.80  =  12400. 

2.  What  amount  of  debt  in  London  can  be  paid  with 
$4000  in  New  York,  rate  of  exchange  as  above  ? 

^  =  -^  =  £833  6s.  8d. 

3.  What  will  a  draft  on  London  for  £480  15s.  6d.  cost 
in  New  York,  rate  as  above? 

15s.     6d. 
12 

■rfo    =  th 
£480|^  X  4.80  =  $2307.72. 

4.  What  will  $500  in  New  York  pay  in  Pans,  exchange 
on  London  as  above,  and  £1  =  25.2  Francs. 


X  ^^-  =  125  X  21  =  2625  Francs. 


5.  What  sum  in  New  York  will  pay  2625  Francs 
in  Paris  ? 

6.  What  will  $2540  in  New  York  pay  in  Paris  ?  as  above. 

7.  What  sum  in  New  York  will  pay  52500  Francs  in 
Paris,  exchange  at  25.2  Francs  to  the  £  and  £1  =  $4.80  p 

Rem. — As  the  rates  of  exchange  are  not  constant,  it  is  not  proper 
to  insert  a  table. 


AvERAGij^G   Accou:n'ts. 


When  sales  are  made  at  different  times  and  on  different 
terms,  to  find  a  mean  time  when  the  whole  may  be  paid 
without  loss  to  either  party ;  thus, 

A  merchant  sells  goods  as  follows : 

January      1st  for  $100  on  1  month. 
January    18th  for  $200  on  1  month. 
February    1st  for  $300  on  2  months. 
*    February  12th  for  $250  on  3  months. 

No  payment  has  yet  been  made.  In  how  many  days 
should  the  whole  be  paid,  in  order  that  there  be  no  lossi 
to  either  party  ? 

Begin  when  the  first  account  is  due,  February  1st. 


$100 

X 

0 

= 

200 

X 

17 

zzz: 

3400 

300 

X 

59 

z=: 

17700 

^-'250 

X 

100 

= 

25000 

850 

850 

)  46100  ( 
4250 
3600 
3400 

54 

days. 

300  =  1^  =  ^. 
Due  54  days  after  February  Isfc,  that  is,  27th  March. 


AVERAGING    ACCOUNTS, 


133 


BANK    ACCOUNT. 

* 

In  bank,  accounts  are  closed  at  the  end  of  the  year, 
and  the  balance,  which  is  always  in  favor  of  the  depositor, 
is  brought  to  the  credit  side  of  the  account,  as  over-draw- 
ing is  not  permitted. 

To  this  balance  each  deposit  is  added  at  the  date  on 
which  it  is  made,  and  from  the  credit  side  is  subtracted 
each  sum  drawn  by  check  at  its  date. 

Each  balance,  or  sum,  is  multiplied  by  the  number  of 
days  from  its  date  until  the  next  transaction,  and  lastly 
by  the  time  between  the  last  transaction  and  the  end  of 
the  year. 

The  sum  of  all  these  products  will  be  the  number  df 
days  that  one  dollar  is  at  interest. 

Dr.    John  Johnson  in  acct.  with  Mechanics'  Bank.    Cr. 


1877. 
Jan.    6. 
"    15. 
"    18. 
"   29. 


1876. 

To  check. 

$200 

Dec.  31. 

«     u 

100 

Jan.  10. 

((          ft 

200 

Jan.  25. 

it           tt 

350 

By  bal.  old  acct. 
By  cash  $300. 
By  cash  $750. 


DA. 

$600 

6 

400 

4 

700 

5 

600 

3 

400 

7 

1150 

4 

800 

2 

3600 
1600 
3500 
1800 
2800 
4600 
1600 


6 )  19,500 
Interest,  3.25 

The  last  sum  or  difference  on  the  credit  side  is  the 
Principal. 

$800 
Interest  =        3.25 

Total  balance  =  $803.25 
The  account  is  only  rendered  for  one  month. 


JLlligatio:n'. 


Alligation  is  the  mixing  of  different  qualities  of 
grain,  groceries,  liquors,  etc.,  either  to  obtain  an  average 
price  of  the  mixture,  which  process  is  termed  Alliga" 
Hon  Medial^  or  to  make  a  mixture  of  several  kinds 
that  shall  have  a  certain  value,  and  this  is  called  J  Hi- 
gation  Alternate. 

ALLIGATION    MEDIAL. 

EXAMPLES. 

1.  Mix  together  10  lbs.  of  tea  at  40  cents  a  IK,  i2  lbs. 
at  50  cts.  a  lb.,  5  lbs.  at  60  cts.  per  lb.,  and  3  1\3S.  <tt  $1.00 
per  lb. ;  what  is  the  mixture  worth  per  lb.  ? 


Jbs.        cts. 

cts. 

10  X  40 

= 

400 

12  X  50 

= 

600 

6  X  60 

= 

300 

^  xlOO 

= 

300 

30  lbs. 

= 

1600    ctfc. 

lib. 

= 

53^  ct«». 

2.  Mix  together  12  gallons  of  wine  at  50  cents  a  gal., 
10  gallons  at  60  cents,  5  gallons  at  80  cents,  and  3  gallons 
of  water;  what  is  the  mixture  worth  per  galloi*? 


ALLIGATION. 

gdl,       cts. 

Cts. 

12  X  50 

=: 

600 

10  X  60 

= 

600 

5  X  80 

m 

400 

_3  X     0 

= 

0 

30  gal. 

nz 

1600  cts. 

Igal. 

=: 

b^  cts.  per  gal. 

135 


3.  Mix  10  gallons  of  wine  worth  $2  a  gallon,  8  gallons 
at  $1.50,  16  gallons  at  $1.25,  and  4  gallons  of  water; 
what  is  one  gallon  worth  ? 


10  X  2 

= 

$20 

8  X  1.50 

zz: 

12 

16  X  1.25 

r= 

20 

J  X  0 

= 

0 

40  gal. 

= 

$52 

1  gal. 

= 

$1.30 

6.  Mix  50  lbs.  coffee  worth  14  cents,  40  lbs.  at  20  cts., 
100  lbs.  at  15  cts.,  and  30  lbs.  at  10  cts.;  what  is  a  lb. 
of  the  mixture  worth  ?  Ans,  15  cts. 

7.  A  farmer  has  10  pigs  worth  $3  each,  12  worth  $4 
each,  and  8  large  ones  worth  $9  each ;  what  is  the  aver- 
age worth  ?  Ans.  $5  each. 

ALLIGATION    ALTERNATE. 

PROBLEM. 

What  relative  quantity  of  each  must  be  taken  of  two 
kinds  of  sugar,  the  one  worth  5  cts.  per  lb.  and  the  other 
9  cts.,  in  order  that  the  mixture  be  worth  6  cts.  per  lb.  ? 


136 


ALLIGATION. 

5 

1 

3      ■  = 

15 

9 

i 

1           = 

41b.  = 

9 
24 

lib.  = 

6 

By  making  1  lb.  at  5  cents  worth  6  cents,  there  is  a 
gain  of  1  cent ;  and  in  1  lb.  at  9  cents  made  at  6  cents, 
there  is  a  loss  of  3  cents;  hence,  in  -J^  of  a  lb.  at  9  cents 
there  is  a  loss  of  1  cent ;  therefore,  in  mixing  1  lb.  at 

5  cents  with  ^  of  a  lb.  at  9  cents,  the  mixture  is  worth 

6  cents,  and  the  gain  and  loss  are  equal.  By  reducing 
the  1  and  ^  to  a  common  denominator  and  multiplying 
both  by  it,  they  become  integral  numbers,  and  tlie  rela- 
tive quantity  of  each  is  the  same, 

OoR.  1. — It  matters  not  how  many  different  qualities 
are  to  be  mixed,  they  must  be  mixed  by  pairs,  one  of  the 
values  less  and  the  other  greater  than  the  mean  value. 

CoE.  2. — If  there  are  odd  numbers  to  be  mixed,  or 
more  above  or  below  the  mean  value,  one  of  those  having 
the  smaller  number  must  be  mixed  with  two  or  more  of 
those  having  the  greater  number  on  the  other  side  of  the 
mean. 

EXAMPLES. 

1.  Mix  coffee  at  15  cts.,  17,  19,  and  22  cts.  per  lb.,  so 
that  the  mixture  is  worth  18  cts. 


18 


15 

19 


18 


1 

1 
3 

18 

t   1 
5   i 

15": 

i 

I 

4 

17-:  i 

1 

A 

1- 

19..;  : 

1 

% 

1 

22- 

i 

I 

3 

18 


15 

22 


18 


15-.- 
17-: 
19-; 
22-^ 


18 


17 
19 

1 
4 
3 
1 


ALLIGATION, 


137 


The  sum  of  these  will  also  prove  a  correct  mixture ; 
that  is, 


5 

at 

15 

=     75 

5 

at 

17 

=     85 

4 

at 

19 

=     76 

4 

at 

%% 

=     88 

18  )  324 

\    18 

CoE. — If  any  two  values,  one  greater  and  the  other  legs, 
be  connected  together,  and  the  difference  between  each 
one  and  the  mean  value  be  placed  opposite  the  one  with 
which  it  is  connected,  the  quantity  opposite  each  value 
will  be  its  relative  quantity. 

Rem. — The  relative  quantity  of  each  is  not  ahsolute,  as  any  two 
opposite  quantities  may  be  mixed,  and,  after  being  mixed  differ- 
ently, these  mixtures  may  be  added  and  the  relative  quantities 
changed. 

2.  A  grocer  has  spices  at  18,  24,  36,  and  42  cts.  per  lb., 
of  which  he  wishes  to  make  a  mixture  worth  32  cents 
per  lb. 


32 


18.... 

10 

= 

180 

24-: 

4 

= 

96 

36-- 

8 

= 

288 

42..-. 

14 

= 

588 

36 

)  1152  (  32  cts. 
108 

72 

■- 

72 

3.  A  merchant  has  five  qualities  of  liquors,  at  the  fol- 
lowing prices  per  gallon,  viz.,  $1.25,  $1.45,  $1.60,  $1.80, 


138 


ALLIGATION. 


and  $1.90;  and  he  has  an  order  for  liquor  at  $1.50  per 
gallon ;  at  the  rate  of  $1.90  he  has  but  40  gallons,  all  of 
which  he  wishes  to  put  in  the  mixture;  how  much  must 
be  taken  of  each  of  the  other  kinds  ? 


150 


$1.25- 
1.45- 
1.60- 
1.80- 

..1.90 


10  +  40  = 

30 

25 

5 

25 

25  X  I  =  40. 


50xf 
30xf 
25  xf 

25  xf 


nz   80 

=  48 
=  40 

:=      8 

=  40 


4.  A  farmer  has  six  kinds  of  wheat  of  different  values 
per  bushel,  viz.,  $1.00,  $1.05,  $1.10,  $1.20,  $1.35,  and 
$1.40;  he  makes  a  sale  at  $1.15  per  bushel;  how  many 
bushels  of  each  kind  must  he  take  ? 


115 


$1.00 

:   25 

=  2500 

1.05  •: 

;  20 

=  2100 

1.10- : : 

i   5 

=   550 

1.20-- : 

;     5 

=   600 

1.35....; 

;    10 

=  1350 

1.40 

•   15 

=  2100 

80 

)  92.00 

1.15,  Proof. 

Note, — Different  connections  will  produce  different  results; 
the  only  principle  involved  is  that  of  the  two  quantities  mixed, 
one  must  be  less  and  the  other  greater  than  the  mean. 


Imvolvtiok  aj{d  EvoLUTioj\r. 


i~ 


Involution  and  JEvolution  correspond  very  nearly 
to  multiplication  and  division.  Involution  consists  in 
the  multiplication  of  the  same  number  by  itself,  or  of  the 
same  factor  employed  two  or  more  times  to  produce  the 
product ;  whilst  Evolution  restores  the  common  factors 
from  the  product. 


INVOLUTION 

is  called  the  raising  of  powers ;  thus,  2  x  2  =  4,  is  called 
the  second  power  of  2,  and  is  expressed  by  2^. 

2x2x2  =  8  =  3d  power  of  2,  and  is  expressed  by  2^. 
3  X  3  =  9  =  2d  power  of  3,  and  is  expressed  by  3^. 

3  X  3  X  3  =  27  =  3d  power  of  3,  and  is  expressed  by  3^. 


THEOREM. 

The  area  of  a  square  is 
equal  to  the  square  of  its 
side. 

As  there  are  12  inches  in 
the  length  of  this  square,  and 
for  every  inch  in  breadth 
there  will  be  12  sq.  in.,  thus, 
12x1=12  sq.  in.;  12x2  = 
24  sq.  in. ;  12  x  5=60  sq.  in,; 
and  12  X  12  =  144. 


140  EVOLUTION, 

The  area  of  the  square  is  equal  to  the  square  of  its  side. 

Cor.  1.— If,  instead  of  taking  the  whole  side  for  its 
breadth,  a  part  only  is  taken,  as  1  in.,  2  in.,  3  in.,  or  5  in., 
the  figure  is  a  rectangle,  and  its  area  is  the  product  of  its 
length  and  breadth. 

CoR.  2. — If  a  diagonal  is  drawn  to 
the  rectangle,  it  divides  it  into  two 
equal  triangles. 

CoR.  3. — It  is  proved  by  a  simple  geometrical  demon- 
stration that  the  area  of  any  parallelogram  is  also  equal 
to  the  product  of  its  base  and  perpendicular  altitude; 
therefore  the  area  of  any  triangle  is  equal  to  one-half  the 
product  of  its  base  and  perpendicular  altitude. 

Rem.  1. — A  plumb-line  which  extends  directly  toward  the  cen- 
tre of  the  earth  is  apparently  perpendicular  to  the  surface  of  the 
earth,  but  really  perpendicular  to  a  plane  which  is  a  tangent  to 
the  surface  of  the  earth. 

Rem.  2. — The  floor  of  a  house  is  either  tangent  to  the  surface  of 
the  earth,  or  is  parallel  to  another  plane  which  is  tangent  to  the 
surface  of  the  earth  * 

Rem.  3. — Planes  holding  these  positions  are  called  horizontal 
planes,  and  planes  which  are  everywhere  equally  distant  are 
parallel 

EVOLUTIOlSr, 

or  the  extracting  of  rootsj  is  exactly  the  reverse  of  invo- 
lution, and  is  expressed  by  the  radical  sign  V  for  square 
root,  and  ^/    for  cube  root. 

As  we  have  seen  in  square  measure  that  the  area  of  the 

*  A  line  touching  a  circle,  or  a  plane  touching  a  sphere  at  but 
one  point  is  called  a  tangent. 


EVOLUTION.  141 

surface  is  found  by  the  product  of  two  equal  sides,  so  in 
evolution  we  have  given  the  surface  to  find  the  side. 

As  the  product  is  the  result  of  two  equal  factors,  it 
appears  to  be  the  natural  solution  to  factor  this  product; 
thus, 

PROBLEMS. 

4  =  2x2;  therefore  2  is  the  square  root  of  4. 
9  =  3x3;  therefore  3  is  the  square  root  of  9. 
16  =  2  X  2  X  2  X  2  =  4  X  4;  .-.  4  is  the  sq.  rootof  16. 


25  =  5x5. 

169  =  13  X 13. 

36  =  ex  6. 

196  =  14  X 14. 

49  =  7x7. 

225  =  15  X 15. 

64  =  8x8. 

256  =  16x16. 

81  =  9x9. 

289  =  17x17. 

100  =  10  X 10. 

324  =  18  X 18. 

121  =  11  X 11. 

361  =  19  X 19. 

144  =  13  X 12. 

400  =  20  X  20. 

Rem. — Above  is  given  all  the  square  numbers  to  400. 

CoE. — For  any  other  numbers  below  400  only  an  ap? 
proximate  root  can  be  obtained. 

2x2x2=  8.  2  is  the  cub6  root  of  8. 
3x3x3=  27.  3  is  the  cube  root  of  27. 
4x4x4=    64.'       4  is  the  cube  root  of  64,  etc. 

5x5x5  =  125.  9  X    9  X    9  =    729. 

6x6x6  =  216.  10  X 10  X 10  =  1000. 

7x7x7  =  343.  11x11x11  =  1331. 

8x8x8  =  512.  12  X  12  X 12  =  1728. 


14^  EVOLUTION. 

Rem. — When  the  given  number  is  not  a  power  of  an  integer, 
the  root  may  still  be  approximated  by  trial  with  decimals.  Another 
method  is  generally  adopted ;  thus, 

You  can  readily  see  the  number  of  figures  the  root  is 
composed  of. 

1x1=  1  9x9=  81 

11 X   11  =       1,21  99  X   99  =       98,01 

111  X  111  =  1,23,21  999  X  999  =  99,80,01 

An  additional  figure  in  the  root  produces  two  in  the 
power.  When  the  left-hand  figure  in  the  root  is  small, 
there  will  be  an  odd  number  of  figures  in  the  power; 
but  when  the  left-hand  figure  is  more  than  3,  there  will 
always  be  twice  as  many  figures  in  the  power  as  are  in 
the  root. 

In  raising  a  number  consisting  of  two  figures  to  the 
second  power;  as, 


12 

12 

144 

10  X  10  = 

100 

2x 

10  X    2  = 

40 

2x2  = 

4 
144 

144  is  composed  of  the  square  of  10,  and  of  twice 
10+2  multiplied  by  2;  twice  10+2  must  therefore  be 
the  second  divisor;  hence. 

10  )  144  ( 

10 

100 
22)44( 

2 

44 

12,  root. 

EVOLUTION,  143 


When  there  axe  three  figures  in 

the  root ;  as, 

123 

123 
369 

246 

123 

16129 

100  X 100  =  10000 

200+20x20=  4400 
(200+40  +  3)3=   729 

100  )  15129  ( 100 

1,51,29  (  123 

10000 

1 

220)  5129  (  20 

22)51 

4400 

44 

243  )  729  (   3 

243 )  729 

729  123 

729 

Rbm. — The  result  of  the  last  process,  without  the  zeros,  is  the 
same  as  of  those  illustrated  with  the  zeros. 

Cor. — To  extract  the  square  root  of  a  number,  point 
it  off  in  periods  of  two  figures,  beginning  with  units. 
Place  in  the  root  the  largest  figure,  the  square  of  which 
is  not  greater,  but  is  either  equal  to,  or  less  than  the  left- 
hand  period  of  the  power.  Square  the  root  obtained,  and 
subtract  the  square  from  the  left-hand  period;  bring 
down  the  next  period,  and  for  a  divisor  double  the  root 
found,  regarding  it  as  tens,  and  try  how  often  it  will  go 
into  the  power  brought  down ;  place  this  figure  in  the 
root,  and  add  it  to  the  divisor,  which  completes  the 
divisor ;  then  divide  as  before,  and  bring  down  the  next 
period ;  again  doubling  the  root,  regarding  it  as  tens,  and 
continue  the  same  process.       w 


144 


EVOLUTION. 


Rem. — As  each  figure  in  the  root  is  placed  to  the  right  of  the 
previous  figure,  each  one  holds  the  place  of  tens  to  the  one  follow- 
ing it. 

There  is  also  a  method  similar  to  this  for  the  extrac- 
tion of  cube  root,  which  I  shall  not  insert  in  this  yolume. 


EXAMPLES. 

1.  Extract  the  square  root  of  441.  Ans.  21. 

2.  Extract  the  square  root  of  9801.  '  Ans.  99. 

3.  Extract  the  square  root  of  15129.  Ans.  123. 

4.  Extract  the  square  root  of  103041.  Ans.  321. 

Rem. — A  fraction  is  squared  when  multiplied  by  itself ;  therefore 
the  root  is  extracted  by  extracting  the  root  of  both  terms  ;  for,  as 

Bj  a  proposition  in  Geometry,  it  is  proved  that  the 
square  described  on  the  hypothenuse  of  a  right-angled 
triangle  is  equivalent  to  the  sum  of  the  squares  described 
on  the  base  and  perpendicular ;  thus. 

Rem. — Two  sides  of  a  square 
are  at  right  angles,  so  also  the 
two  sides  of  a  rectangle  ;  as  the 
sides  of  a  house,  the  two  sides 
forming  a  corner  of  the  house 
are  at  right  angles;  the  two 
sides  of  the  carpenter's  square. 

In  this  figure,  the  base  is 
regarded  as  3  inches,  the  per- 
pendicular 4,  and  the  hypothe- 
nuse 5 ;  and 

[(3x3)=9]  +  [(4x4)=16]=35 

and  5  X  5  =  25. 


^ 


EVOLUTION. 


145 


Cor. — Any  two  sides  of  a  right-angled  triangle  being 
given,  the  third  side  can  be  found ;  for  if  the  base  and 
perpendicular  are  given,  add  their  squares,  and  the  square 
root  of  the  sum  will  be  the  hypothenuse  ;  if  the  hypoth- 
enuse  and  one  side  are  given,  square  both,  subtract  their 
squares,  and  the  square  root  of  the  difference  will  be  the 
other  side. 

Rem. — Tlie  figure  ABC  is  exemplified  by  the  wall  of  a  house, 
which  is  always  at  right  angles  with  the  surface  of  the  earth,  and 
a  ladder  at  the  distance  of  the  base  from  the  foot  of  the  wall, 
peaching  the  wall  at  the  height  of  the  perpendicular.  AB  is  the 
base,  AC  the  perpendicular,  and  BC  the  hypothenuse. 


EXAMPLES. 

1.  What  is  the  length  of  the  diagonal  of  a 
square,  each  side  of  which  is  9  feet. 

Rem. — AC  is  the  diagonal  corresponding  to  the 
hypothenuse  of  a  right-angled  triangle. 


9    X 
9   X 


=     81 

=.     81 


162  (12.72  +  ,  Diagonal. 
1 


22)  62 
44  • 

247  )  1800 
1729 


2542  )  7100 
5084 

2016 


146  EVOLUTION, 

Cor.  1. — As  the  angles  B  and  D  are  both  right  angles, 
hence  equal,  if  the  one  is  placed  on  the  other  they  will 
coincide;  and  as  the  sides  AB,  AD,  DC,  and  BC,  are 
equal,  these  sides  will  fall  upon  each  other  and  coincide ; 
und  AC  is  common  to  the  two  triangles  ABC  and  ADC ; 
tlierefore  the  two  triangles  are  equal,  and  each  is  one-half 
the  square. 

CoR.  2. — A  triangle  is  one-half  a  square  or  rectangle, 
and  the  surface  of  a  square  or  rectangle  is  equal  to  the 
product  of  its  two  sides,  or  of  the  base  and  perpendicu- 
lar; therefore  the  surface  of  a  triangle  is  one-half  the 
product  of  its  base  and  altitude. 

ScHO. — A  triangle  that  is  not  right-angled,  is  one-half 
a  parallelogram;  and  it  is  proved  in  Geometry  that  a 
rectangle  and  a  parallelogram  of  the  same  base  and  alti- 
tude are  equivalent ;  hence  the  above  holds  good  for  any 
triangle. 

2.  A  ladder  15  feet  long,  placed  9  feet  from  the  foot  of 
the  wall  of  a  house,  reaches  the  window  of  the  second 
story  of  the  house ;  how  high  is  the  window  ? 

15  X  15  =  225 

9x9=  81 

Vlil  =  12  feet. 

3.  How  many  square  feet  in  the  floor  or  ceiling  of  f^ 
room  20  ft.  long  and  15  ft,  wide  ? 

20  X  15   =r  300  sq.  ft. 

4.  How  many  square  feet  in  the  wall  of  a  room  20  ft 
long  and  9  ft.  high  ?  Ans.  180  sq.  ft. 

5.  How  many  square  feet  in  a  wall  15  ft.  wide  and 
9  ft.  high  ?  Ans.  135  sq.  ft. 


EVOLUTION.  147 

6.  How  many  square  feet  in  the  four  walls  of  a  room 
20  ft.  long,  15  ft.  wide,  and  9  ft.  high  ?   Ans.  630  sq.  ffc. 

7.  How  many  square  feet  of  plastering,  including  the 
ceiling  ?  Ans.  930  sq.  ft. 

8.  How  many  square  feet  in  a  triangle  whose  base  is 
15  ft.  and  altitude  8  ft.  ? 

i(15  X  8)  =  60  sq.ft. 

9.  How  many  cubic  feet  in  a  cubical  block,  the  side  of 
each  square  face  being  5  feet  ? 

5  X  5  X  5  =  125  cu.  ft. 

10.  How  many  cubic  feet  in  the  room  above,  20  by  15 
and  9  high  ? 

Rem.— The  area  of  a  circle  is  found  by  squaring  the  diameter 
and  multiplying  the  square  by  .7854. — "Geometry." 

11.  What  is  the  number  of  square  feet  in  a  circle 
whose  diameter  is  4  ft.  ? 

4  X  4  X  .7854  —  12.5664  sq.  ft. 

12.  How  many  cubic  feet  in  a  cylinder  whose  circular 
base  is  2  feet  in  diameter  and  its  altitude  3  feet  ? 

2  X  2  X  .7854  x  3  = 

13.  How  many  cubic  feet  in  a  pyramid  whose  base  is 
3  feet  square  and  altitude  9  feet  ? 

i  (3  X  3  X  9)  =  27  cu.  ft. 

14.  How  many  cubic  feet  in  a  cone  whose  circular  base 
is  3  feet  in  diameter  and  altitude  9  feet  ? 

i  (3  X  3  X  .7854  x  9). 

Note. — The  volume  of  a  pyramid  and  also  of  a  cone  is  the  area 
of  the  base  multiplied  by  one-third  the  altitude. 


\ 


Series  of  Commok  Bifferemceb. 


A  Series  of  Common  Differences^  usually 
termed  Arithmetical  Progression,  is  one  in  which  the 
difference  of  any  two  consecutive  terms,  taken  in  order, 
is  the  same  ;  as,  in  the  series  1,  2,  3,  4,  5,  etc.,  the  first 
term  is  1,  and  the  common  difference  is  1 ;  and  it  is  an 
increasing  series  as  each  successive  term  is  greater  than 
the  preceding  one. 

1,  3,  5,  7,  etc.,  is  an  increasing  series,  and  the  common 
difference  is  2. 

2,  5,  8,  11,  etc.,  is  an  increasing  series,  and  the  common 
difference  is  3. 

12,  10,  8,  6,  4,  2,  etc.,  is  a  decreasing  series,  and  the 
common  difference  is  2. 

THEOREM     I. 

The  sum  of  a  series  of  equal  differenees  is  equal 
to  one-half  the  product  of  the  number  of  terms  and 
the  sum  of  the  first  and  last  terms. 

As,  the  sum  of  six  terms  of  the  series,.  1,  2,  3,  4,  etc.,  is 

12        3        4        5        6 
6        5        4        3        2        1 


SERIES    OF    COMMON   DIFFERENCES, 


149 


Sum  of  6  terms  of  both  series  is 

6    X   7     =     42. 
Sum  of  6  terms  of  one  series  is 

1(6   X   7)     =     21. 

The  sum  of  8  terms  of  the  series : 

2    5    8   11   14   17 
23   20   17   14   11    8 


20 
5 


23 

2 


25   25   25   25   25   25   25   25 

The  sum  of  2  series  is 

8   X   25     =     200 
The  sum  of  1  series  is 

^(8   X   25)     =     100 

Therefore  the  sum  of  a  series  of  common  differences  is 
one-half  the  product  of  the  number  of  terms,  multiplied 
by  the  sum  of  the  first  and  last  terms. 

THEOREM     II. 

Any  term  of  an  increasing  series  is  equal  to  the 
first  term  plus  the  common  difference  taken  as 
many  times  as  the  number  of  the  terms  less  one ; 
and  any  terin  of  a  decreasing  series  is  equal  to  the 
first  term  minus  the  quantity  that  is  to  be  added 
above. 

The  series  may  be  written 

12  3  4  5 

a,    a  +  d,    a-\-2d,    a  +  Sd,    a  +  4td,  etc., 
and        a,    a—d,    a— 2d,    a^M,    a— 4 J,  etc., 

in  which  a  represents  the  first  term  and  d  the  common 
difference,  and  in  each  term  d  is  once  less  than  the  num- 


150  SERIES    OF    COMMON   DIFFERENCES, 

ber  of  terms,  either  plus  or  minus ;  hence  any  term  of  a 
series  of  equal  differences,  is  equal  to  the  first  term  plus 
or  minus  the  common  difference  taken  as  many  times  as 
the  number  of  the  term  less  one. 

Let  n  represent  the  number  of  terms,  and  any  term 
may  be  regarded  as  the  last  term,  and  let  I  represent  the 
last  term  ]  thus, 

I  z=z  a  +  {n  —  1)  d'y 
and  I  z=z  a  —  {n  —  1)  d, 

EXAMPLES. 

1.  What  is  the  sum  of  25  terms  of  the  series  1,  2,  3, 
4,  5,  etc.  ? 

S^^  ^l^2h)  =  325. 

2.  What  is  the  10th  term  of  the  series  1,  3,  5,  etc.  ? 

10th  term  ^=  I  =z  a  -\-  {n  —  1)  d. 
Z  =  1  +  18  =  19. 

3.  What  is  the  sum  of  the  10  terms  of  the  series  ? 

S  z=zi^{l  +  19)  =  100. 

4.  Find  the  6th  term  of  the  series  12, 10,  8,  etc. 

U\i  =  l  =  a—  {n  —  l)d. 
Z  =r  12  —  10  =  2. 

5.  Find  the  sum  of  six  terms  of  the  series. 

f  (12  +  2)  =  42. 

6.  Find  the  sum   of  100  terms  of  the  series  1,  2, 

3,  4,  etc. 

I  =  100. 
^  =  loii  (1  _!_  100)  =  5050. 


Gej^ebal    Review. 


In  the  preceding  pages  almost  every  yariety  of  Arith- 
metical Principles  has  been  analyzed  and  developed,  and 
quite  a  number  of  examples  have  been  given  under  each 
head.  The  following  Practical  Examples  have  been 
selected  in  order  to  give  the  pupil  a  general  review  of  the 
course.  He  will  find  it  beneficial  to  refer  back  to  the 
analyses,  as  he  can  never  be  successful  in  practice  with- 
out a  thorough  knowledge  of  principles.  Let  this  be  his 
motto :  "  To  gain  a  knowledge  of  the  principle  first,  and 
then  solve  examples  under  the  principle  until  it  is  indeli- 
bly fixed  in  the  mind. 

PRACTICAL      EXAMPLES. 

1.  The  divisor  is  21,  the  quotient  35,  and  the  remainder 
12 ;  what  is  the  dividend  ?  Arts.  747. 

2.  The  three  factors  are  25,  36,  and  42 ;  what  is  the 
number?  Ans.  37,800. 

3.  Eeduce  37,800  to  its  prime  factors. 

Ans.  5,  5,  2,  2,  3,  3,  2,  3,  7. 

4.  Find  the  greatest  common  divisor  of  12, 18,  24,  30. 

Ans,  6. 

5.  Find  the  least  common  multiple  of  5,  15,  18,  36. 

Arts.  180. 

6.  Find  the  least  common  denominator  of  -^,  -^,  -^,  ^. 

Ans.  180. 

7.  Find  the  least  common  denominator  of  ^,  \,  I,  ^,  -J-. 

Ans.  60. 


152  PRACTICAL   EXAMPLES. 

8.  If  I  of  a  ton  of  coal  cost  $2,  what  will  5^;^  tons  cost? 

Arts.   135.50. 

9.  If  A  can  dig  a  ditch  in  9  days,  B  in  12  days,  and  C 
in  15  days,  how  long  would  it  take  the  three  men  if  they 
work  together?  Arts.  Sff  days. 

10.  A  merchant  bought  goods  for  $4,675,  on  which  he 
wishes  to  gain  20^  ;  how  much  must  he  sell  the  goods 
for?  Ans,  85,610. 

11.  In  a  cargo  of  damaged  goods  a  merchant  has  goods 
which  cost  $7,420,  and  on  which  he  is  obliged  to  lose 
25^  ;  what  must  the  sales  amount  to  ?      A7is,  $5,565. 

12.  A  ship  caught  in  a  storm  was  obliged  to  throw 
overboard  one-fourth  of  its  cargo.  At  what  rate  per  cent, 
of  gain  must  a  merchant  sell  his  goods  in  order  that  he  may 
neither  gain  nor  lose  by  the  transaction?     Ans.  33^^. 

13.  A  ship  on  a  cruise  has  provisions  to  last  1,200  men 
6  months;  at  the  end  of  4  months  finds  that  they  cannot 
return  in  less  than  6  more  months ;  how  many  men  must 
disembark,  in  order  that  those  remaining  may  have  pro- 
visions for  that  time  ? 

Ans.  800  men  must  leave  the  vessel. 

14.  What  cost  6|  yards  of  cloth  at  $2|  a  yard? 

Ans.  $18. 

15.  At  $1  per  bushel,  how  many  bushels  of  apples  can 
be  bought  for  $18  ?  Ans,  24  bush. 

16.  How  much  land  at  $37|^  per  acre  can  be  bought 
for  $625  ?  '  Ans.  16f  acres. 

17.  The  product  of  two  factors  is  f ,  and  one  of  the  fac- 
tors is  f ;  what  is  the  other  factor?  Ans.  ^. 

18.  The  quotient  is  -~  and  the  dividend  | ;  what  is  the 
divisor?  Ans.  f^. 


PRACTICAL  EXA21FLES,  153 

19.  If  8  horses  cost  $900,  what  will  12  horses  cost  ? 

Ans.  $1,350. 
""'    20.  If  f  of  an  acre  of  land  cost  $120,  what  will  llj 

acres  cost  ?  Ans,  $1,575. 

f^  21.  A  farmer  sold  to  a  merchant  5  cows  at  $24.50  each, 
2  horses  at  $76.25  each,  and  12  sheep  at  $4.75  each ;  in 
payment  he  received  28  yards  of  carpet  at  $1.25  per  yard, 
36  yards  cloth  at  $4.25  per  yard,  and  the  balance  in 
money ;  how  much  money  did  he  receive  ?   Ans,  $144. 

22.  Washington  was  born  22d  Feb.,  1732,  and  died 
14th  Dec.  1799 ;  what  was  his  age  ? 

Ans.  67  y.  9  m.  22  d. 

23.  John  Adams  was  born  Oct.  19th,  1735,  and  died 
July  4th,  1826 ;  what  was  his  age  ?  Ans,  90  y.  8  m.  15  d. 

24.  Thomas  Jefferson  was  born  April  2d,  1743,  and 
died  July  4th,  1826  ;  how  old  was  he  ? 

Ans,  83  y.  3  m.  2  d. 

25.  A  note  for  $300,  due  three  months  after  date,  was 
given  April  1st,  1875,  and  the  principal  and  interest  at  Q% 
was  paid  Oct.  16th,  1876 ;  what  was  the  amount  ? 

Ans,  $323.25. 

26.  If  3  men  build  2:^  rods  of  wall  in  one  day,  how 
many  rods  will  6  men  build  in  8  days  ?    Ans,  ^^  rods. 

27.  If  5  men  in  4  days  build  12  rods,  how  many  men 
will  build  72  rods  in  15  days.  Ans,  8  men. 

28.  What  is  the  cost  of  collecting  $24,000  taxes  at  ^%  ? 

A71S,  $120.' 

29.  What  is  the  premium  on  a  bond  for  $3,000  at  |^  ? 

Ans,  $22.50. 

30.  What  is  the  commission  on  $3,600  at  l^%  ? 

Ans,  $54. 


154  PRACTICAL   EXAMPLES, 

31.  A  hare  is  120  rods  ahead  of  a  greyhound,  and  the 
hare  runs  8  rods  while  the  hound  runs  12 ;  how  many 
rods  will  the  hound  run  until  he  overtakes  the  hare  ? 

Ans.  360  rods. 

32.  A  and  B  bought  a  horse  for  $40 ;  A  paid  $25  and 
B  $15 ;  they  afterward  sold  the  horse  for  $64 ;  how  much 
should  each  receive  ?  Ayis,  A  $40  and  B  $24. 

.  33.  A  has  $12,  B  has  $15,  C  $18,  and  D  $24;  what  part 
of  the  whole  has  each  one  ? 

Ans.  A,  ^ ;  B,  ^;  0,  ^,  and  D  ^. 

34.  What  is  the  cost  of  15  cwt.  2  qr.  15  lbs.  sugar  at  6 
cts.  per  lb.?  Ans.  $93.90. 

35.  What  per  cent,  is  ^?  Ans.  50%. 
86.  What  per  cent,  is  i  ?  Ans.  26%. 

What  per  cent,  is  ^  ?  A7is.  33  J^.     ■ 

What  per  cent,  is  f  ?  Arts.  66f  ^. 

What  per  cent,  is  I"?  Ans.  16^%. 

What  per  cent,  is  ^  ?  Ans.  14f  ^. 

What  per  cent,  is  f  ?  Ans.  28^^^. 

Eem.  One-half  of  if  ^  is  ^^-^,  or  J  of  100;^  is  50^ ;  one- 
fourth  o^iU  is  tVo.  or  i  of  100^  is  25^;  H^,  or  100^, 
is  equal  to  1  or  a  whole. 

37.  What  is  the  interest  of  $240  for  5  years  at  6^  ? 

Ans.  $72. 

38.  What  is  the  interest  of  $634.50  for  3  years  at  6%  ? 

Ans.  $114.21. 

39.  What  is  the  interest  of  $760  for  4  years  at  5%  ? 

Ans.  $152. 

40.  What  is  the  interest  of  $836  for  3^  years  at  4^^  ? 

Ans.  $131.67. 


PRACTICAL   EXAMPLES.  155 

41.  What  is  the  interest  of  $556  for  8  months  at  6^  ? 

.    Ans,  122.24. 

42.  What  is  the  interest  of  $864  for  9  months  at  Q%  ? 

Ans.  $38.88. 

43.  What  is  the  interest  of  $8,425  for  16  months  at  Q%  ? 

Ans.  $674. 
Kem.  At  6^  for  any  number  of  months,  the  rate  %  is 
one-half  the  number  of  months. 

44.  A  and  B  traded  together.  A  put  in  $1,200  for  16 
months,  and  B  $1,500  for  12  months;  they  gained  $500; 
how  much  of  the  gain  should  each  receive  ? 

Ans.  A,  $258-^;  B,  $241ff. 

45.  A  farmer  being  asked  how  many  sheep  he  had,  re- 
plied that  they  were  in  four  fields ;  in  the  first  -J  of  the 
whole,  in  the  second  J,  in  the  third  ^,  and  in  the  fourth 
150  sheep;  how  many  had  he  ?  Ans.  600  sheep. 

46.  If  I  buy  coffee  at  16  cts.  per  lb.  and  sell  it  for  20 
cts.,  what  ^  do  I  gain  ?  Ans.  2b%. 

47.  What  is  the  present  value  of  a  note  for  $300,  due  in 
8  months,  bank  discount,  interest  worth  6^,  without 
grace?  Ans.  $288.     • 

48.  What  are  the  proceeds  of  a  note  for  $500,  due  in 
3  months,  discounted  in  bank,  3  days'  grace  ? 

Ans.  $492.25. 

49.  The  proceeds  of  a  note  of  $3,000  for  4  months,  6^, 
3  days'  grace,  are  ?  $2,938.50. 

50.  What  is  the  bank  discount  of  a  note  for  $2,000,  at 
90  days,  6^  ?  Ans.  $31. 

51.  A  man  spends  f  of  his  income  for  board,  f  of  the 
remainder  for  clothing,  and  has  $30  left;  what  is  his  in- 
come? Ans.  $270. 


156  PRACTICAL   EXAMPLES. 

52.  If  a  man  travel  39  mi.  7  fur.  8  rd.  in  7  hours,  what 
is  the  rate  per  hour  ?  Ans.  5  mi.  5  fur.  24  rd. 

53.  The  sum  of  two  numbers  is  ^,  and  their  difference 
|-;  what  are- the  numbers?  Ans.  ^  and  ff. 

54.  If  a  quantity  of  provisions  serve  1,200   men  12* 
months,  how  long  will  three  times  as  many  provisions 
serve  2,400  men  ?  Ans.l^  months. 

55.  What  will  12  A.  3  K.  20  P.  of  land  cost  at  $30  per 
acre  ?  Ans,  12|  x  $30  =  8386.25. 

56.  What  cost  35  A.  2  E.  10  P.  at  $20  per  acre  ? 

35^2^  A.    Am.  $711.25. 

57.  A,  B,  and  C  together  can  do  a  piece  of  work  in  a 
certain  time.  A  and  B  can  do  it  in  y\  of  the  time,  and 
B  and  0  can  do  it  in  ^  of  the  time.  In  what  time  can 
each  one  do  it  alone  ?    Ans.  K  in  y\,  B  in  y^,  C  in  -^. 

58.  A  can  do  a  piece  of  work  in  -^  of  a  day,  B  in  \,  and 
C  in  ^ ;  how  long  will  it  take  all  working  together  ? 

Ans.  1^. 

59.  How  much  grain  must  I  take  to  mill,  so  as  to  have 
nine  bushels  remaining  after  the  miller  has  taken  one- 

.  tenth  toll  ?  Ans.  10  bu. 

60.  For  what  sum  must  I  give  my  note,  in  bank,  at  90 
days,  rate  6^,  in  order  to  receive  three  hundred  dollars  ? 

Int.  glfo.  ^300  =  iMf ;  .-.  $300  X  fin  =  ^304.72. 

61.  By  selling  goods  for  $936.45  I  lost  10^;  had  I 
gained  10^,  what  would  I  have  sold  them  for  ? 

Eate-i^.    Ans.  $1,144.55. 

62.  I  have  fifty  gallons  of  wine,  worth  $1.50  per  gallon ; 
how  much  water  must  I  add  to  reduce  the  price  25  cts. 
per  gallon  ?  Ans.  10  gals. 

63.  I  bought  40  yds.  of  cloth  at  5^  less  than  cost,  and 


PRACTICAL  EXAMPLES.  157 

sold  it  at  10%  above  first  cost,  and  gained  $15;  what  was 
the  first  cost  per  yard  ? 

Ans,  ^=$15,  i^=$100,  VoP-=l^-50. 

Eem.  Two  terms  form  a  ratio,  and  two  equal  ratios  a 
proportion;  thus, 

64.  As  3  is  to  4,  so  is  6  to   ?  Ans,  8. 

65.  As  5  is  to  7,  so  is  15  to   ?  Ans,  21. 

66.  As  I  is  to  f,  so  is  I  to  ?  Ans,  1. 

67.  As  I  is  to  I,  so  is  H  to  ?  Ans.  If. 

68.  As  |-  is  to  I,  so  is  I-  to   ?  Ans.  ^. 

Eem.  1.  In  any  proportion  the  odd  terms  are  antece- 
dents, and  the  even  terms  consequents ;  each  antecedent 
and  consequent  is  termed  a  couplet ;  when  one  or  more 
of  the  terms  of  a  couplet  are  fractional,  both  terms  must 
be  reduced  to  a  common  denominator ;  and  the  numera- 
tors will  express  the  ratio. 

Eem.  2.  A  complete  proportion  has  four  terms,  and  if 
each  consequent  is  made  the  numerator  of  a  fraction  and 
each  antecedent  a  denominator,  the  fractions  are  equal, 
and  may  be  put  in  the  form  of  an  equation;  thus,  if 
3  :  4  : :  6  :  8,  then  f =f ;  they  will  also  form  an  equation 
by  alternating  the  terms;  thus,  f =f. 
Let  these  equations,  i=  I  ^^'^    f=  f 

be  cleared  of  fractions  by  multiplying,  24=24  24=24 
each  numerator  by  the  other  denominator;  thus,  and 
here  we  find  that  the  product  of  the  first  and  fourth, 
called  the  extremes,  is  equal  to  the  product  of  the  second 
and  third,  called  the  means,  and  by  this  principle,  any 
three  of  the  terms  being  known,  the  fourth  is  readily 
found. 


158  PRACTICAL  EXAMPLES. 

CoE.  As  the  antecedents  aud  consequents  may  be  alter- 
nated, it  is  best  to  make  the  unknown  the  last  term. 

Instead  of  as  3  is  to  4,  so  is  6  :  8,  they  may  be  written 
as4:  3  ::  8:  6,  or8:  6::4:  3. 

69.  If  3  acres  of  grass  yield  9  tons  of  hay,  how  many 
tons  will  7  acres  yield  ? 

3 

Ans.  3  :  7  : :  9  :  ^^=^1  tons. 

CoE.  As  the  product  of  the  means  and  of  the  extremes 
are  equal,  the  fourth  term  will  be  the  quotient  derived 
from  the  product  of  the  means  divided  by  the  one  ex- 
treme. 


70.  Given  |=| 
5     r 

3:5::6:? 

iJA=10. 

„.        3      ? 
Given  -=.^ 

5 : 3  : :  10  :  ? 

^1-^=6. 

Given  ^=A 

10  :  6  : :  5  :  ? 

^=3. 

,,.        3      6 

Given  ^=^ 

6  :  10  : :  3 : ? 

H''J=5. 

CoE.  Either  antecedent  is  to  its  consequent  as  the 
other  antecedent  is  to  its  consequent ;  or,  either  conse- 
quent is  to  its  antecedent  as  the  other  consequent  is  to  its 
antecedent. 

If  three  men  mow  8  acres  of  grass  in  4  days,  how  many 
acres  will  6  men  mow  in  3  days  ? 

A71S,  3  men  in  4  days  is  equal  to  12  men  in  1  day ; 
and  6  men  in  3  days  is  equal  to  18  men  in  1  day ;  hence 
1^:18::  8:  ^41^  ==12  acres. 

71.  A  can  mow  1  acre  in  a  day,  B  1^  acres,  and  C  2 


PRACTICAL   EXAMPLES,  159 

acres;  in  how  many  days  will  they  mow  18  acres  all 
-working  together  ?  Arts,  4  days. 

72.  A  can  do  a  piece  of  work  in  2|-  days,  B  in  3^  days; 
how  long  will  they  take  working  together  ? 

Ans.  If  days. 

73.  A  garrison  of  960  men  have  provisions  to  last  8 
months;  at  the  end  of  3  months  400  men  leave;  how 
long  will  the  provisions  last  those  remaining  ? 

Ans,  8f  months. 

74.  If  by  selling  tea  at  12^^  profit,  I  gain  10  cts.  per 
lb.,  what  did  the  tea  cost  per  lb.  ? 

121^=^10,100^=80.     ^7Z5.  80cts. 

75.  Bought  a  lot  of  bacon  at  10  cts.  per  lb.;  the  loss  in 
weight  is  10^ ;  what  must  I  sell  it  at  per  lb.  to  gain  20^  ? 

ioxYxif§-=-4^.:=13J.     A71S.  13|  cents. 
Eem.  The  cost  of  -^  lb.  is  10  cents. 

76.  I  commenced  business  with  $1,200  and  gained  20^ 
each  year  on  my  stock  ;  the  gain  was  added  to  the  stock 
each  year ;  at  the  end  of  three  years  I  closed  with  ? 

si2ooxi||xi2|xi||^|2,073.6.    Ans,  12,073.60. 

77.  The  interest  on  $750  at  %%  was  '$150;  how  long 
had  the  note  been  due?  J^s^— J^.     Ans,  3  y.  4  m. 

"'^8.  A  man  dying  leaves  ^  of  his  property  to  his  wife, 
/f  of  the  remainder  to  his  son,  |-  what  remained  to  one 
daughter,  and  the  balance,  $5,000,  to  another  daughter; 
what  was  the  share  of  each  ? 

Ans,  The  wife,  $15,000,  the  son  $20,000,  and  daughter 
$5,000. 

79.  Paid  50  cts.  premium  for  gold  at  ^%  above  par ; 
how  much  gold  did  I  get  ? 

1^=50  cts.;  then  l^=:5ox|_g2j  cts.,  and  100^=the 
whole.    Ans.  $62.50. 


160  PRACTICAL  EXAMPLES. 

80.  If  13  men  mow  a  field  in  21  days,  in  what  time 
would  39  men  mow  it?  Ans.  7  days. 

81.  What  is  the  difference  between  21-J  and  20|  ? 

Ans,  1\. 

82.  Eeduce  |  x  |  to  a  simple  fraction.  Ans,  ff . 

T      "6" 

4  A  ^ 

83.  Eeduce  to  a  simple  fraction  1-^4-  Ans.  ^, 

1_L.|.      4—1 

84.  Eeduce  | — j  x  , — v  to  a  simple  fraction. 

85.  Eeduce  i  X  |  x  |  x  f -^f  x|x|xftoa  simple  frac- 
tion. A71S,  -^. 

oa    T>  A        8  X 12  X 15  X 18  X  21  X  24  X  30  ,  .      . 

86.  Eeduce ^^7; — ^^ — jtz — ts to  a  simple 

20  X  36  X  42  X  48  ^ 

fraction.  Ans,  270. 

87.  The  product  of  two  numbers  is  ^\,  and  one  of  the 
numbers  is  f ;  what  is  the  other  number  ?         Ans,  f. 

88.  The  minuend  is  y\,  and  the  remainder  is  f ;  what 
is  the  subtrahend  ?  Ans,  -^. 

89.  f  is  how  many  times  f  ?  1^  ?  A  ?  A  ?  A?  ^9  ? 
A?  A? 

90.  I  is  how  many  times  ,2^?  ^?  ^  ?  ^  ?  ^?  ^  ? 

A?  A? 

91.  How  many  bottles  of  1  qt.  1  pt.  each  can  be  filled 
from  a  hogshead  (63  gals.)  of  wine  ?  Ans.  168  bottles. 

92.  How  many  tiles  6  in.  by  4  will  cover  a  house  50  ft. 
long,  and  rafters  on  each  side  24  ft.  long  ? 

Ans.  14,400  shingles. 

93.  How  many  square  yards  of  carpeting  will  cover  a 
room  48  ft.  long  and  30  feet  wide  ?  Ans,  160  yds. 


PRACTICAL  EXAMPLES.  161 

94.  How  many  cubic  yards  of  excavation  in  a  mass  of 
earth  90  ft.  long,  30  ft.  wide,  and  24  ft.  deep. 

Arts.  2,400  cu.  yd. 

95.  What  is  the  cost  of  1  T.  5  cwt.  65  lbs.  beef  at  8  cts. 
per  lb.?  2,565  1b.    Ans.  $205.20. 

96.  How  many  cords  of  wood  in  a  pile  28  ft.  long,  8  ft. 
high,  and  4  ft.  wide  ?  Ans,  7  cords. 

97.  How  many  acres  in  a  rectangular  tract  of  land  480 
rods  long  and  60  rods  wide  ?.  A7is.  180  acres. 

98.  At  25  cents  a  gallon,  how  many  gallons  of  beer  can 
be  bought  for  $57.25  ?  Ans.  229  gallons. 

99.  Keduce  1,276  farthings  to  the  higher  denomina- 
tion. £1  6s.  7d. 

100.  The  discount  of  a  note  at  90  days,  6^,  was  $31 ; 
Hhat  was  the  face  of  the  note  given  ? 

^3^=:$31,  and  31  ^^f^= the  whole.    Ans.  $2,000. 

101.  A  note  with  interest  for  4  years  3  months,  at  6%, 

was  paid;  the  amount  was  $1,255  ;  what  was  the  face  of 

the  note  ? 

f  ^      25|^      51  .     .     ,       ,  .   ,       ^251     ,^^^ 

l00'^200'^P^^^^P  mterest=^-=1255; 

126  6X||^^      ^/15.   $1,000. 

102.  A  farm  of  375  A.  2  R.  and  24  P.  was  sold  at 
$51.50  per  acre ;  what  did  the  sale  amount  to  ? 

3751^  A.     Ans.  $19,345.97f 

103.  How  many  rotations  will  a  wheel  12  ft.  in  circum- 
ference make  in  a  mile  ?  Ans.  440  rotations. 

104.  How  many  barrels,  each  containing  2  bu.  3  pk., 
can  be  filled  with  275  bu.  of  apples  ?        Ans,  100  bar. 

105.  What  is  the  difference  between  25  lb.  8  oz.  9  dr. 
and  12  lb.  9  oz.  10  dr.  of  lead  ?  Ans.  12  lb.  14  oz.  15  dr. 


162  PRACTICAL  EXA3IPLES. 

106.  What  is  the  sum  of  12  bu.  3  pk.  5  qts.  wheat  and 
25  bu.  2  pk.  4  qt.  Ans.  38  bu.  2  pk.  1  qt. 

107.  Find  the  two  equal  factors  4,  9, 16,  25,  36,  49,  64, 
81,  100,  121,  144,  169,  196,  225,  256,  289,  324,  361,  400, 
441,  484. 

Eem.  One  of  two  equal  factors  is  the  square  root  of  the 
number,  and  one  of  three  equal  factors  is  the  cube  root. 

108.  Find  one  of  the  three  equal  factors  of  8,  27,  64, 
125,  216,  1,000. 

109.  How  many  seconds  in  12  years  of  365  days  5  h. 
48  m.  48  sec.  Ans,  378,683,136  sec. 

110.  How  many  suits  of  clothing  can  be  made  from 
169  yds.,  each  suit  having  6  yds.  2  qrs.?  Ans.  26  suits. 

111.  How  many  minutes  in  31  days  ? 

Ans.  44,640  min. 

112.  A  man's  income  in  2  years  was  $2,333^,  and  the 
second  year  was  33^^  more  than  the  first.  What  was  it 
each  year?  Ans.  1st,  $1,000;  2d,  $l,333f 

113.  Muslin  bought  at  6  cts.  and  sold  at  7^  cts.;  what 
is  the  gain  per  cent.?  Ans.  25^. 

114.  K  receives  $126  interest  on  a  bond,  rate  1% ;  what 
is  the  face  of  the  bond  ?  l^^l^^.    Ans,  $1,800. 

115.  On  a  note  of  $800,  A  received  $300,  which  was 
the  interest  for  five  years ;  what  was  the  rate  ? 

*^A=$60  for  1  year;  rate  WV=Jq^-    ^^5-  Hfc- 

116.  On  a  note  of  $300,  at  6%,  he  received  $76.50  in- 
terest; what  was  the  time?         -^^.     ^^5.  4  y.  3  m. 

117.  In  w^hat  time  will  $600  produce  $120  interest  at 
6%  ?  ^.     Ans,  3  y.  4  m. 

118.  In  what  time  will  $500  at  6%  amount  to  $650  ? 

J^.     Ans.  5  years. 


PRACTICAL  EXAMPLES,  163 

119.  For  what  sum  must  a  note  be  drawn  for  60  days, 
at  6^,  to  net  $3,000  ?        «3 o  o  o  >'^^,    Ans.  $3,031.83. 

120.  A  man  owes  $300,  due  in  4  months,  $500  in  6  m., 
$1,000  in  7  m.  24  days,  and  $1,200  in  5  m.  In  what  time 
may  it  all  be  paid  without  loss  or  gain  ? 

Ans.  6  months. 

121.  Sold  goods  as  follows: 

April  10th,  $300  on  3  m. 

May   20th,  $480  on  4  m. 

June  30th,  $600  on  2  m. 
At  what  time  may  the  whole  be  paid  without  gain  or 
loss.  Ans.  August  26th. 

122.  A  lot  is  32  rods  in  length,  and  contains  one  acre ; 
how  wide  is  it?  Ans,  5  rods. 

123.  What  is  the  interest  of  $486,  at  6%,  from  April 
16th,  1873,  to  July  1st,  1876  ?  A71S.  $93.55|. 

124.  What  is  the  interest  of  $1,200,  at  6%,  from  April 
1st,  1874,  t6  Jan'y  21st,  1877  ?    What  is  the  amount  ? 

Ans.  $202  Int.,  $1,402  Amt. 

125.  What  is  the  interest  of  $3,600,  at  6%,  for  2  years 
3  m.  and  10  days  ? 

2  y.  =  720  days 
3m.=  90  days 

10  days  ^ 

"820  days    .-.    •'a«^^x,^=r$492. 
Sem.  As  a  general  rule  it  is  simpler  and  shorter  to  re- 
duce the  time  to  days. 

126.  What  is  the  interest  of  $6,000,  at  6%,  for  3  years 
5  m.  25  days?  Ans.  $1,255. 

Observe  that  the  interest  of  $6,000  is  $1  a  day,  reckon- 
ing 360  days  to  the  year. 


164  PRACTICAL  EXAMPLES. 

137.  What  is  the  interest  of  $4,000  fpr  636  days  ? 

4xt;36.    Ans,  $424. 

128.  What  is  the  interest  of  $1,000  for  1,248  days  ? 

ix  1,248.    Ans.  $208. 

129.  What  is  the  interest  of  $2,000  for  624  days  ? 

Ans.  $208. 

130.  What  is  the  interest  of  $60  for  3,000  days  ? 

Ans,  $30. 

131.  What  is  the  interest  of  $742  for  3  y.  7  m.  24  d.? 

Ans,  $162,498. 

132.  What  is  the  cost  of  12  gal.  3  qt.  1  pt.  wine  at  $3.00 
per  gal.?  12|.    Ans.  $38|. 

133.  What  cost  2  bu.  3  pk.  4  qt.  apples  at  80  cts.  per 
bushel  ?  2|  bu.    Ans,  $2.30. 

134.  The  product  is  8f,  and  one  of  the  factors  is  3f ; 
what  is  the  other  factor  ?  "  Ans,  2^. 

135.  The  dividend  is  12|,  and  the  quotient  5f ;  what  is 
the  divisor  ?  Ans.  2 J. 

136.  The  product  of  three  factors  is  4,125,  and  one 
factor  is  15  ;  what  is  the  product  of  the  other  two  ? 

Ans.  $275. 

137.  Two  men  engage  a  piece  of  work ;  the  one  can  do 
it  in  5  days,  and  the  other  in  7^  days ;  how  long  will  it 
take  both  together  ?  Ans.  3  days. 

138.  A  tradesman  hired  a  journeyman  at  24  pence  for- 
every  day  he  worked,  and  he  was  to  forfeit  6d.  for  every 
idle  day.    At  the  end  of  30  days  his  account  balanced ; 
how  many  days  did  he  work  ?  Ans.  6  days. 


ANSWERS   NOT    GIVEN 

IN    THE    PEECEDING    PAGES, 


AbBITIOK    AJ^B    SUBTBACTlOJf. 

Page  16. 

1.    115  sheep.  2.     145  fowls. 

3.  63  in  first  and  21  in  second. 

Page  17. 

4.  45  cattle  in  third  field.  9.  256  verses. 

5.  28  cents  left.  10.  190  cents. 

6.  139  pages  to  read.  11.  200  dollars. 

7.  155  eggs.  12.  18830  dollars. 

8.  46  chickens.  13.  13843  dollars  left,  i 

Page  18. 

14.    14000  dollars.  15.    1193  acres  left. 

16.  888  dollars  left. 

17.  893  acres  unsold,  247  acres  of  second  farm,  and 
64C  acres  of  third  farm. 


166 


DIVISION. 


18.  Sold  for  467  doUars.         20.     Gained  $1182. 

19.  Gained  1118  dollars.        21.     $3779. 


MULTIPLICATIOJV. 

Page  26. 

1.  18796796.    4.  17600100.    7.  1913247450. 

2.  12895164.    5.  505489116.   8.  8585904128. 

3.  32671178.    6.  3515971700.  9.  982259375. 

10.    984381300. 


Divi  sioj^. 


I*affe  31. 

1 

Quo.    1915,  rena 

L.  98. 

7.  Quo 

800368,  rem.    209. 

2. 

u 

1477,    « 

60. 

8.     « 

373792,    «       979. 

3. 

iC 

263,    « 

234 

9.     " 

439169,    «      1458 

4. 

cc 

5475,    " 

116. 

10.    « 

13482,    «         42. 

5. 

6C 

209205,    « 

992. 

11.     " 

4457,    «  102170, 

6. 

U 

780317,    « 

300. 

12.    " 

676,    "196432. 

1. 

50  cts. 

4. 

10  lbs. 

7.    200  cts. 

3. 

5  lbs. 

5. 

150  cents. 

8.     20  lbs. 

3. 

100  cents. 

6. 
10. 

15  lbs. 
25  lbs. 

9.     250  cts. 

Page  32. 

11. 

1960  cents. 

13. 

$15660. 

15.    1207680. 

13. 

245  lbs. 

14. 

348  acres. 

16.    3245  acres. 

FACTORING.  167 

17.  $1875.  19.     $2625.  21.     $3780. 

18.  15  horses.         20.    35  oxen.  22.    84  cows. 

Page  33. 

32.  Horses  cost  $2016,  cows  $1944,  sheep  $1456,  all 
cost  $5416. 

33.  $5740.  34.     Cost  $76600,  sold  for  $75355. 

35.  Gained  $6912. 

Page  34. 

36.  Paid  $25  per  acre,  and  sold  for  $29  per  acre. 

37.  Paid  $65  per  acre,  sold  for  $71. 

38.  Each  cow  cost  $40. 

39.  Isf  Ans.,  36000  farms;    2d  Ans.,  184000  farms; 
Srd  Ans.,  1028000  farms. 

40.  1st  State,  $144,000,000 ;  2d  State,  $588,800,000 ; 
3rd  State,  $822,400,000. 

41.  Each  daughter's  share  $7100. 


Factoeij^g 


Page 

50. 

1. 

60  =  3,  %,  3,  5. 

9. 

84  =  3,  3,  3,  7. 

3. 

64  =  3,  3,  2,  3,  3,  3. 

10. 

86  =  3,  43. 

3. 

65  =  5, 13; 

11. 

88  =  3,  3,  3, 11. 

4. 

70  =  2,  5,  7. 

13. 

90  =  3,  3,  3,  5. 

5. 

73  =  3,  3,  3,  3,  3. 

13. 

95.  =  5, 19. 

6. 

75  =  3,  5,  5. 

14. 

96  =  3,  3,  3,  3,  3,  3. 

7. 

78  =  2,  3, 13. 

15. 

98  =  3,  7,  7. 

8.    80  =  2,  2,  2,  2,  5.  16.  100  =  2,  2,  5,  5. 


58 

FB ACTIONS. 

17. 

102  =  2,  3, 17. 

27. 

124  =  2,  2,  31. 

18. 

104  =  2,  2,  2, 13. 

28. 

125  =  5,  5,  5. 

19. 

106  =  2,  53. 

29. 

128  =  2,2,2,2,2,2,3. 

20. 

108  =  2,  2,  3,  3,  3. 

30. 

130  =  2,  5, 13. 

21. 

110  =  2,  5, 11. 

31. 

132  =  2,  2,  3, 11. 

22. 

112  =  2,  2,  2,  2,  7. 

32. 

136  =  2,  2,  2, 17. 

23. 

114  =  2,  3, 19. 

33. 

140  =  2,  2,  5,  7. 

24. 

116  =  2,  2,  29. 

34. 

225  =  5,  5,  9. 

25. 

118  =  2,  59. 

35. 

500  =  2,  2, 5,  5,  5. 

26. 

120  =  2,  2,  2,  3,  5. 

36. 

625  =  5,  5,  5,  5. 

Multiplication  of  Fbactioj^s. 

Page  63. 

7.  i^  X  4  =  1^  =  58|. 

8.  ^  X  6  =  ^  =  103f 

9.  ^  X  9  =  ^  =  164}. 

2 

10.  -^  X  $  =  198. 

11.  22204.  12.     64916|. 

13.  3147  X  J^^  =  ^^V^^   =  112505I-. 

532 

14.  i6|^  X  ^H^  =   16627  X  532  =  8845564. 

15.  -H*^  X  ^  =  ^H^  =  1711H- 

16.  isj,  X  -V-  =  ^-  =  143f|. 

17.  ij^  X  W  =  ^WP  =  380fff. 

18.  ^  X  H^  =  -^W^  =  2139ff 


DENOMINATE   NUMBERS.  169 


DirisiOM  OF  Fbactioj^s. 

Page  65. 

L    /^  X  I  =  45.  4.    I  X  I  =  A- 

2.  *^  X  I  =  18.  5.    i  X  I  =  |. 

3.  «0  X  J  =  38.  6.    f  X  f  =  f  :«  l\. 
7.    ^  X  ^  =  W  =  5A- 


8. 

4x4  =  1  =  1*. 

5 

9. 

H^  X  A  =  -W  =  5A. 

10. 

15  X  Jf  =  ^  =  32^. 

11. 

^  X  ^  =  W-  =  85V. 

12. 

H  X  J  =  W  =  2M- 

13. 

H  X  V-  =  «i  =  ItIs- 

14. 

^F  X  3^  =  Vs^  =  7^. 

15. 

^  X  ^  =  fiH  =  SHU- 

Page  75. 

.     1.    f  of  T^  =  ^;  .-.  i  =  150,000,  and  the  whole  is 
eight  times  I  =  $400,000. 

2,    They  will  meet  in  7^\  hours,  A  will  have  traveled 
45|^  miles,  and  B  61-^  miles. 

Dej^omij^ate  J^umbers. 

Page  90. 

4.     ^291    155.    9d. 


170  DENOMINATE    NUMBERS, 


Page  91. 

2. 

Ans.,  500  cents ;  5000  mills. 

3. 

700  cents ;  7000  mills. 

4. 

515  cents ;  5150  mills.           5.    6153  mills. 

Page  92. 

9. 

63  dollars  25  cents  7  mills;  $63,257. 

10. 

$753.25.            11.    $9.00.            12.    $10.50. 

13. 

$105.                 14.    $1,125.          15.    $39,375. 

16. 

$55.65. 

17. 

Cost  $483.136 ;  sold  for  $640,328. 

Page  93. 

24. 

9303  far.                      26.    £39   19s.  3d.  3  far. 

25. 

£39442  Ifar.             27.    £1  19s.  2d.  ^tss. 

Page  102. 

1. 

3359  far.                      7.    1705577  dr. 

2. 

3068d.                   8.    1  T.  6  cwt.  1  qr  1  lb,  11  oz, 

3. 

1682d.                           9.    32871  qr. 

4. 

£2  9s.  2  far.               10.    11  lb.  2  oz.  4  qr. 

5. 

£2  5s.  3d.                    11.    14690  gr. 

6. 

£36  lis.                    12.    lib.  21  13  2  gr. 

13. 

741402  in. 

14. 

4  Leagues  5  fur.  12  rods  4  yd.  1  ft.  4  in. 

15. 

177i  inches.               18.    162  inches. 

16. 

135  inches.                 19.    172048  sq.  in. 

17. 

180  inches.                 20.    161940  cu.  in. 

RATIO.  171 

Page  103. 

22.     831giUs.  23.    633  pts. 

JPage  106. 

56.  1440  yds. ;  1200  yds. ;  960  yds. ;  640  yds. ;  480  yds.; 
384  yds.;  320  yds.;  274f  yds. 

57.  Proceeds  83312.50  and  3785f  yds. 

Ratio. 

JPage  113. 

5.     16  days.  6.     120  feet. 

Page  114. 

7.    $25. 

9.  $10.50;  $11.40;  $28.50;  $38;  $57;  $47;  $70.50 
$117;  $128.10. 

10.  $11.87^;    $12.81i;    $19.50;    $46.50;    $119.50 
$157;  $188.50;  $349.25. 

11.  $260;    $310;   $500;    $525;  $690;   $735;  $780 
$825  ;  $1510  ;  $1995. 

12.  $37.50;    $25;    $12.50;    $10;    $20;    $30;    $40 
$60;  $70. 

13.  $lf|.  14.    $9.  15.    $18.75  and  $22.50. 

Page  116. 

3.    Arts.,  9 


172  EVOLUTION. 

PEBCEJfTAGE. 

Page  131. 

5.    $500.  6.    13335  Francs.  7.    $10,000 

EVOLUTIOJr. 

Page  147. 

10.    2700cti.ft.  13.    9.4248  cu.  ft. 

14    21.2058  cu.  ft. 


THIS  BOOK  IS  DUE  ON  THK  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OP  25  CENTS 

WILL  BE  ASSESSED   FOR   FAILURE  TO   RETURN 
THIS   BOOK   ON   THE   DATE   DUE.    THE   PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY    AND    TO    $1.00    ON    THE    SEVENTH     DAY 
OVERDUE. 

FEB  19  1942  F 

I" 


YB   17349 


■■^ 


^ 

^ 
J 


911326 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


\ 


BAKE  R'  S 

nVt^^HOVED    AND    SIMPLIFIED 

SRIES  OF  MiT":.^^!FICS. 


By  ANDREW   H.    BA' 


mw^^'^^ 


".USt 


Pibor. 


L 


'•le  .us  m.. 
a  the  pn  ^.c^.'l 
ysis,  thor-    igbi,^>^ 
brief  and  ex; 'licit, 

iie  books  of  tb-  .  eries  <-| 
I      PRIMARY  ARltjJMT 
11.     ELEMENTARY 
III.     COMPLETE     A^ 
ITL-  r^^QMETRY 
.GEBRA  i 
WIC 


a*  1 


